Time{Independent Perturbation Theoryweb.mit.edu/6.732/www/new_appendices.pdf† Cohen-Tannoudji, Diu and Lalo˜e, Quantum Mechanics, vol. 2, Ch. 11. † T-Y. Wu, Quantum Mechanics,

Appendix A

Time–Independent Perturbation

Theory

References

• Davydov - Quantum Mechanics, Ch. 7.

• Morse and Feshbach, Methods of Theoretical Physics, Ch. 9.

• Shankar, Principles of Quantum Mechanics, Ch. 17.

• Cohen-Tannoudji, Diu and Laloe, Quantum Mechanics, vol. 2, Ch. 11.

• T-Y. Wu, Quantum Mechanics, Ch. 6.

A.1 Introduction

Another review topic that we discuss here is time–independent perturbation theory becauseof its importance in experimental solid state physics in general and transport properties inparticular.

There are many mathematical problems that occur in nature that cannot be solved ex-actly. It also happens frequently that a related problem can be solved exactly. Perturbationtheory gives us a method for relating the problem that can be solved exactly to the onethat cannot. This occurrence is more general than quantum mechanics –many problems inelectromagnetic theory are handled by the techniques of perturbation theory. In this coursehowever, we will think mostly about quantum mechanical systems, as occur typically insolid state physics.

Suppose that the Hamiltonian for our system can be written as

H = H0 + H′ (A.1)

where H0 is the part that we can solve exactly and H′ is the part that we cannot solve.Provided that H′ ¿ H0 we can use perturbation theory; that is, we consider the solution ofthe unperturbed Hamiltonian H0 and then calculate the effect of the perturbation Hamil-tonian H′. For example, we can solve the hydrogen atom energy levels exactly, but whenwe apply an electric or a magnetic field, we can no longer solve the problem exactly. For

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this reason, we treat the effect of the external fields as a perturbation, provided that theenergy associated with these fields is small:

H =p2

2m− e2

r− e~r · ~E = H0 + H′ (A.2)

where

H0 =p2

2m− e2

r(A.3)

and

H′ = −e~r · ~E. (A.4)

As another illustration of an application of perturbation theory, consider a weak periodicpotential in a solid. We can calculate the free electron energy levels (empty lattice) exactly.We would like to relate the weak potential situation to the empty lattice problem, and thiscan be done by considering the weak periodic potential as a perturbation.

A.1.1 Non-degenerate Perturbation Theory

In non-degenerate perturbation theory we want to solve Schrodinger’s equation

Hψn = Enψn (A.5)

where

H = H0 + H′ (A.6)

and

H′ ¿ H0. (A.7)

It is then assumed that the solutions to the unperturbed problem

H0ψ0n = E0

nψ0n (A.8)

are known, in which we have labeled the unperturbed energy by E0n and the unperturbed

wave function by ψ0n. By non-degenerate we mean that there is only one eigenfunction ψ0

n

associated with each eigenvalue E0n.

The wave functions ψ0n form a complete orthonormal set

∫

ψ∗0n ψ0

md3r = 〈ψ0n|ψ0

m〉 = δnm. (A.9)

Since H′ is small, the wave functions for the total problem ψn do not differ greatly from thewave functions ψ0

n for the unperturbed problem. So we expand ψn′ in terms of the completeset of ψ0

n functions

ψn′ =∑

n

anψ0n. (A.10)

Such an expansion can always be made; that is no approximation. We then substitute theexpansion of Eq.A.10 into Schrodinger’s equation (Eq. A.5) to obtain

Hψn′ =∑

n

an(H0 + H′)ψ0n =

∑

n

an(E0n + H′)ψ0

n = En′

∑

n

anψ0n (A.11)

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and therefore we can write∑

n

an(En′ − E0n)ψ0

n =∑

n

anH′ψ0n. (A.12)

If we are looking for the perturbation to the level m, then we multiply Eq. A.12 from the leftby ψ0∗

m and integrate over all space. On the left hand side of Eq.A.12 we get 〈ψ0m|ψ0

n〉 = δmn

while on the right hand side we have the matrix element of the perturbation Hamiltoniantaken between the unperturbed states:

am(En′ − E0m) =

∑

n

an〈ψ0m|H′|ψ0

n〉 ≡∑

n

anH′mn (A.13)

where we have written the indicated matrix element as H′mn. Equation A.13 is an iterative

equation on the an coefficients, where each am coefficient is related to a complete set of an

coefficients by the relation

am =1

En′ − E0m

∑

n

an〈ψ0m|H′|ψ0

n〉 =1

En′ − E0m

∑

n

anH′mn (A.14)

in which the summation includes the n = n′ and m terms. We can rewrite Eq.A.14 toinvolve terms in the sum n 6= m

am(En′ − E0m) = amH′

mm +∑

n6=m

anH′mn (A.15)

so that the coefficient am is related to all the other an coefficients by:

am =1

En′ − E0m −H′

mm

∑

n6=m

anH′mn (A.16)

where n′ is an index denoting the energy of the state we are seeking. The equation (A.16)written as

am(En′ − E0m −H′

mm) =∑

n6=m

anH′mn (A.17)

is an identity in the an coefficients. If the perturbation is small then En′ is very close to E0m

and the first order corrections are found by setting the coefficient on the right hand sideequal to zero and n′ = m. The next order of approximation is found by substituting for an

on the right hand side of Eq. A.17 and substituting for an the expression

an =1

En′ − E0n −H′

nn

∑

n′′ 6=n

an′′H′nn′′ (A.18)

which is obtained from Eq.A.16 by the transcription m → n and n → n′′. In the above, theenergy level En′ = Em is the level for which we are calculating the perturbation. We nowlook for the am term in the sum

∑

n′′ 6=n an′′H′nn′′ of Eq.A.18 and bring it to the left hand

side of Eq. A.17. If we are satisfied with our solutions, we end the perturbation calculationat this point. If we are not satisfied, we substitute for the an′′ coefficients in Eq. A.18 usingthe same basic equation as Eq.A.18 to obtain a triple sum. We then select out the am term,bring it to the left hand side of Eq.A.17, etc. This procedure gives us an easy recipe to findthe energy in Eq.A.11 to any order of perturbation theory. We now write these iterationsdown more explicitly for first and second order perturbation theory.

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1st Order Perturbation Theory

In this case, no iterations of Eq.A.17 are needed and the sum∑

n6=m anH′mn on the right

hand side of Eq. A.17 is neglected, for the reason that if the perturbation is small, ψn′ ∼ ψ0n.

Hence only am in Eq. A.10 contributes significantly. We merely write En′ = Em to obtain:

am(Em − E0m −H′

mm) = 0. (A.19)

Since the am coefficients are arbitrary coefficients, this relation must hold for all am so that

(Em − E0m −H′

mm) = 0 (A.20)

or

Em = E0m + H′

mm. (A.21)

We write Eq.A.21 even more explicitly so that the energy for state m for the perturbedproblem Em is related to the unperturbed energy E0

m by

Em = E0m + 〈ψ0

m|H′|ψ0m〉 (A.22)

where the indicated diagonal matrix element of H′ can be integrated as the average of theperturbation in the state ψ0

m. The wave functions to lowest order are not changed

ψm = ψ0m. (A.23)

2nd order perturbation theory

If we carry out the perturbation theory to the next order of approximation, one furtheriteration of Eq.A.17 is required:

am(Em − E0m −H′

mm) =∑

n6=m

1

Em − E0n −H′

nn

∑

n′′ 6=n

an′′H′nn′′H′

mn (A.24)

in which we have substituted for the an coefficient in Eq. A.17 using the iteration relationgiven by Eq.A.18. We now pick out the term on the right hand side of Eq. A.24 for whichn′′ = m and bring that term to the left hand side of Eq.A.24. If no further iteration is to bedone, we throw away what is left on the right hand side of Eq. A.24 and get an expressionfor the arbitrary am coefficients

am

[

(Em − E0m −H′

mm) −∑

n6=m

H′nmH′

mn

Em − E0n −H′

nn

]

= 0. (A.25)

Since am is arbitrary, the term in square brackets in Eq.A.25 vanishes and the second ordercorrection to the energy results:

Em = E0m + H′

mm +∑

n6=m

|H′mn|2

Em − E0n −H′

nn

(A.26)

in which the sum on states n 6= m represents the 2nd order correction.

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To this order in perturbation theory we must also consider corrections to the wavefunction

ψm =∑

n

anψ0n = ψ0

m +∑

n6=m

anψ0n (A.27)

in which ψ0m is the large term and the correction terms appear as a sum over all the other

states n 6= m. In handling the correction term, we look for the an coefficients, which fromEq.A.18 are given by

an =1

E′n − E0

n −H′nn

∑

n′′ 6=n

an′′H′nn′′ . (A.28)

If we only wish to include the lowest order correction terms, we will take only the mostimportant term, i.e., n′′ = m, and we will also use the relation am = 1 in this order ofapproximation. Again using the identification n′ = m, we obtain

an =H′

nm

Em − E0n −H′

nn

(A.29)

and

ψm = ψ0m +

∑

n6=m

H′nmψ0

n

Em − E0n −H′

nn

. (A.30)

For homework, you should do the next iteration to get 3rd order perturbation theory, inorder to see if you really have mastered the technique (this will be an optional homeworkproblem).

Now look at the results for the energy Em (Eq. A.26) and the wave function ψm (Eq. A.30)for the 2nd order perturbation theory and observe that these solutions are implicit solu-tions. That is, the correction terms are themselves dependent on Em. To obtain an explicitsolution, we can do one of two things at this point.

1. We can ignore the fact that the energies differ from their unperturbed values in cal-culating the correction terms. This is known as Raleigh-Schrodinger perturbationtheory. This is the usual perturbation theory given in Quantum Mechanics texts andfor homework you may review the proof as given in these texts.

2. We can take account of the fact that Em differs from E0m by calculating the correction

terms by an iteration procedure; the first time around, you substitute for Em thevalue that comes out of 1st order perturbation theory. We then calculate the secondorder correction to get Em. We next take this Em value to compute the new secondorder correction term etc. until a convergent value for Em is reached. This iterativeprocedure is what is used in Brillouin–Wigner perturbation theory and is a better ap-proximation than Raleigh-Schrodinger perturbation theory to both the wave functionand the energy eigenvalue for the same order in perturbation theory.

The Brillouin–Wigner method is often used for practical problems in solids. For example, ifyou have a 2-level system, the Brillouin–Wigner perturbation theory to second order givesan exact result, whereas Rayleigh–Schrodinger perturbation theory must be carried out toinfinite order.

Let us summarize these ideas. If you have to compute only a small correction by per-turbation theory, then it is advantageous to use Rayleigh-Schrodinger perturbation theory

185

because it is much easier to use, since no iteration is needed. If one wants to do a moreconvergent perturbation theory (i.e., obtain a better answer to the same order in perturba-tion theory), then it is advantageous to use Brillouin–Wigner perturbation theory. Thereare other types of perturbation theory that are even more convergent and harder to usethan Brillouin–Wigner perturbation theory (see Morse and Feshbach vol. 2). But these twotypes are the most important methods used in solid state physics today.

For your convenience we summarize here the results of the second–order non–degenerateRayleigh-Schrodinger perturbation theory:

Em = E0m + H′

mm +′∑

n

|H′nm|2

E0m − E0

n

+ ... (A.31)

ψm = ψ0m +

′∑

n

H′nmψ0

n

E0m − E0

n

+ (A.32)

where the sums in Eqs.A.31 and A.32 denoted by primes exclude the m = n term. Thus,Brillouin–Wigner perturbation theory (Eqs.A.26 and A.30) contains contributions in secondorder which occur in higher order in the Rayleigh-Schrodinger form. In practice, Brillouin–Wigner perturbation theory is useful when the perturbation term is too large to be handledconveniently by Rayleigh–Schrodinger perturbation theory, but still small enough for per-turbation theory to work insofar as the perturbation expansion forms a convergent series.

A.1.2 Degenerate Perturbation Theory

It often happens that a number of quantum mechanical levels have the same or nearly thesame energy. If they have exactly the same energy, we know that we can make any linearcombination of these states that we like and get a new eigenstate also with the same energy.In the case of degenerate states, we have to do perturbation theory a little differently, asdescribed in the following section.

Suppose that we have an f -fold degeneracy (or near-degeneracy) of energy levelsψ0

1, ψ02, ...ψ

0f

︸ ︷︷ ︸

states with the same or nearly the same energy

ψ0f+1, ψ

0f+2, ....

︸ ︷︷ ︸

states with quite different energies

We will call the set of states with the same (or approximately the same) energy a“nearly degenerate set” (NDS). In the case of degenerate sets, the iterative Eq. A.17 stillholds. The only difference is that for the degenerate case we solve for the perturbed energiesby a different technique, as described below.

Starting with Eq.A.17, we now bring to the left-hand side of the iterative equation allterms involving the f energy levels that are in the NDS. If we wish to calculate an energywithin the NDS in the presence of a perturbation, we consider all the an’s within the NDSas large, and those outside the set as small. To first order in perturbation theory, we ignorethe coupling to terms outside the NDS and we get f linear homogeneous equations in thean’s where n = 1, 2, ...f . We thus obtain the following equations from Eq. A.17:

a1(E01 + H′

11 − E) +a2H′12 +... +afH′

1f = 0

a1H′21 +a2(E

02 + H′

22 − E) +... +afH′2f = 0

......

. . ....

a1H′f1 +a2H′

f2 + . . . +af (E0f + H′

ff − E) = 0.

(A.33)

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In order to have a solution of these f linear equations, we demand that the coefficientdeterminant vanish:

∣∣∣∣∣∣∣∣∣∣

(E01 + H′

11 − E) H′12 H′

13 . . . H′1f

H′21 (E0

2 + H′22 − E) H′

23 . . . H′2f

......

.... . .

...H′

f1 H′f2 . . . . . . (E0

f + H′ff − E)

∣∣∣∣∣∣∣∣∣∣

= 0 (A.34)

The f eigenvalues that we are looking for are the eigenvalues of the matrix in Eq.A.34 andthe set of orthogonal states are the corresponding eigenvectors. Remember that the matrixelements H′

ij that occur in the above determinant are taken between the unperturbed statesin the NDS.

The generalization to second order degenerate perturbation theory is immediate. In thiscase, Eqs. A.33 and A.34 have additional terms. For example, the first relation in Eq.A.33would then become

a1(E01 + H′

11 − E) + a2H′12 + a3H′

13 + . . . + afH′1f = −

∑

n6=NDS

anH′1n (A.35)

and for the an in the sum in Eq. A.35, which are now small (because they are outside theNDS), we would use our iterative form

an =1

E − E0n −H′

nn

∑

m6=n

amH′nm. (A.36)

But we must only consider the terms in the above sum which are large; these terms areall in the NDS. This argument shows that every term on the left side of Eq. A.35 will havea correction term. For example the correction term to a general coefficient ai will look asfollows:

aiH′1i + ai

∑

n6=NDS

H′1nH′

ni

E − E0n −H′

nn

(A.37)

where the first term is the original term from 1st order degenerate perturbation theoryand the term from states outside the NDS gives the 2nd order correction terms. So, ifwe are doing higher order degenerate perturbation theory, we write for each entry in thesecular equation the appropriate correction terms (Eq. A.37) that are obtained from theseiterations. For example, in 2nd order degenerate perturbation theory, the (1,1) entry to thematrix in Eq.A.34 would be

E01 + H′

11 +∑

n6=NDS

|H′1n|2

E − E0n −H′

nn

− E. (A.38)

As a further illustration let us write down the (1,2) entry:

H′12 +

∑

n6=NDS

H′1nH′

n2

E − E0n −H′

nn

. (A.39)

Again we have an implicit dependence of the 2nd order term in Eqs.A.38 and A.39 on theenergy eigenvalue that we are looking for. To do 2nd order degenerate perturbation we again

187

have two options. If we take the energy E in Eqs.A.38 and A.39 as the unperturbed energyin computing the correction terms, we have 2nd order degenerate Rayleigh-Schrodingerperturbation theory. On the other hand, if we iterate to get the best correction term, thenwe call it Brillouin–Wigner perturbation theory.

How do we know in an actual problem when to use degenerate 1st or degenerate 2ndorder perturbation theory? If the matrix elements H′

ij coupling members of the NDS vanish,then we must go to 2nd order. Generally speaking, the first order terms will be much largerthan the 2nd order terms, provided that there is no symmetry reason for the first orderterms to vanish.

Let us explain this further. By the matrix element H′12 we mean (ψ0

1|H′|ψ02). Suppose

the perturbation Hamiltonian H′ under consideration is due to an electric field ~E

H′ = −e~r · ~E (A.40)

where e~r is the dipole moment of our system. If now we consider the effect of inversionon H′, we see that ~r changes sign under inversion (x, y, z) → −(x, y, z), i.e., ~r is an oddfunction. Suppose that we are considering the energy levels of the hydrogen atom in thepresence of an electric field. We have s states (even), p states (odd), d states (even), etc.The electric dipole moment will only couple an even state to an odd state because of theoddness of the dipole moment under inversion. Hence there is no effect in 1st order non–degenerate perturbation theory for situations where the first order matrix element vanishes.For the n = 1 level, there is, however, an effect due to the electric field in second order sothat the correction to the energy level goes as the square of the electric field, i.e., | ~E|2. Forthe n =2 levels, we treat them in degenerate perturbation theory because the 2s and 2pstates are degenerate in the simple treatment of the hydrogen atom. Here, first order termsonly appear in entries coupling s and p states. To get corrections which split the p levelsamong themselves, we must go to 2nd order degenerate perturbation theory.

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Appendix B

1D Graphite: Carbon Nanotubes

In this appendix we show how the tight binding approximation (§B.1.1) can be used toobtain an excellent approximation for the electronic structure of carbon nanotubes whichare a one dimensional form of graphite obtained by rolling up a single sheet of graphiteinto a seamless cylinder. In this appendix the structure and the electronic properties ofa single atomic sheet of 2D graphite and then discuss how this is rolled up into a cylin-der, then describing the structure and properties of the nanotube using the tight bindingapproximation.

B.1 Structure of 2D graphite

Graphite is a three-dimensional (3D) layered hexagonal lattice of carbon atoms. A singlelayer of graphite, forms a two-dimensional (2D) material, called 2D graphite or a graphenelayer. Even in 3D graphite, the interaction between two adjacent layers is very smallcompared with intra-layer interactions, and the electronic structure of 2D graphite is a firstapproximation of that for 3D graphite.

In Fig. B.1 we show (a) the unit cell and (b) the Brillouin zone of two-dimensionalgraphite as a dotted rhombus and shaded hexagon, respectively, where ~a1 and ~a2 are unitvectors in real space, and ~b1 and ~b2 are reciprocal lattice vectors. In the x, y coordinatesshown in Fig. B.1, the real space unit vectors ~a1 and ~a2 of the hexagonal lattice are expressedas

~a1 =

(√3

2a,

a

2

)

, ~a2 =

(√3

2a,−a

2

)

, (B.1)

where a = |~a1| = |~a2| = 1.42 ×√

3 = 2.46A is the lattice constant of two-dimensionalgraphite. Correspondingly the unit vectors ~b1 and ~b2 of the reciprocal lattice are given by:

~b1 =

(2π√3a

,2π

a

)

, ~b2 =

(2π√3a

,−2π

a

)

(B.2)

corresponding to a lattice constant of 4π/√

3a in reciprocal space.Three σ bonds for 2D graphite hybridize in a sp2 configuration, while, and the other

2pz orbital, which is perpendicular to the graphene plane, makes π covalent bonds. InSect. B.1.1 we consider only the π energy bands for 2D graphite, because we know that theπ energy bands are covalent and are the most important for determining the solid stateproperties of 2D graphite.

189

y

k

x

k

y

x

a

2a

1

(a) (b)

BAΓ

K

M

2

b

b

1

Figure B.1: (a) The unit cell and (b) Brillouin zone of two-dimensional graphite are shownas the dotted rhombus and shaded hexagon, respectively. ~ai, and ~bi, (i = 1, 2) are unitvectors and reciprocal lattice vectors, respectively. Energy dispersion relations are obtainedalong the perimeter of the dotted triangle connecting the high symmetry points, Γ, K andM .

B.1.1 Tight Binding approximation for the π Bands of Two-Dimensional

Graphite

Two Bloch functions, constructed from atomic orbitals for the two inequivalent carbonatoms at A and B in Fig. B.1, provide the basis functions for 2D graphite. When weconsider only nearest-neighbor interactions, then there is only an integration over a singleatom in the diagonal matix elements HAA and HBB, as is shown in Eq. 1.81 and thusHAA = HBB = ε2p. For the off-diagonal matrix element HAB, we must consider the three

nearest-neighbor B atoms relative to an A atom, which are denoted by the vectors ~R1, ~R2,and ~R3. We then consider the contribution to Eq. 1.82 from ~R1, ~R2, and ~R3 as follows:

HAB = t(ei~k·~R1 + ei~k·~R2 + ei~k·~R3)

= tf(k)(B.3)

where t is given by Eq. 1.831 and f(k) is a function of the sum of the phase factors of ei~k·~Rj

(j = 1, · · · , 3). Using the x, y coordinates of Fig. B.1(a), f(k) is given by:

f(k) = eikxa/√

3 + 2e−ikxa/2√

3 cos

(kya

2

)

. (B.4)

Since f(k) is a complex function, and the Hamiltonian forms a Hermitian matrix, we writeHBA = H∗

AB in which ∗ denotes the complex conjugate. Using Eq. (B.4), the overlap integralmatrix is given by SAA= SBB = 1, and SAB = sf(k) = S∗

BA. Here s has the same definition

1We often use the symbol γ0 = |t| for the nearest neighbor transfer integral.

190

K Γ M K-10.0

-5.0

0.0

5.0

10.0

15.0

Ene

rgy

[eV

] π∗

ππ

π∗

K K

E

KMΓKM

[eV

]

Figure B.2: The energy dispersion relations for 2D graphite are shown throughout thewhole region of the Brillouin zone. Here we use the parameters ε2p = 0, t = −3.033eVand s = 0.129. The inset shows the electronic energy dispersion along the high symmetrydirections of the triangle ΓMK shown in Fig. B.1(b) (see text).

as in Eq. 1.84, so that the explicit forms for H and S can be written as:

H =

ε2p tf(k)

tf(k)∗ ε2p

, S =

1 sf(k)

sf(k)∗ 1

. (B.5)

By solving the secular equation det(H−ES) = 0 and using H and S as given in Eq. (B.5),the eigenvalues E(~k) are obtained as a function w(~k), kx and ky:

Eg2D(~k) =ε2p ± tw(~k)

1 ± sw(~k), (B.6)

where the + signs in the numerator and denominator go together giving the bonding πenergy band, and likewise for the − signs, which give the anti-bonding π∗ band, while thefunction w(~k) is given by:

w(~k) =√

|f(~k)|2 =

√

1 + 4 cos

√3kxa

2cos

kya

2+ 4 cos2

kya

2. (B.7)

In Fig. B.2, the energy dispersion relations of two-dimensional graphite are shown through-out the 2D Brillouin zone and the inset shows the energy dispersion relations along the highsymmetry axes along the perimeter of the triangle shown in Fig. B.1(b). The upper half ofthe energy dispersion curves describes the π∗-energy anti-bonding band, and the lower halfis the π-energy bonding band. The upper π∗ band and the lower π band are degenerateat the K points through which the Fermi energy passes. Since there are two π electronsper unit cell, these two π electrons fully occupy the lower π band. Since a detailed calcu-lation of the density of states shows that the density of states at the Fermi level is zero,two-dimensional graphite is a zero-gap semiconductor.

191

When the overlap integral s becomes zero, the π and π∗ bands become symmetricalaround E = ε2p which can be understood from Eq. (B.6). The energy dispersion relationsin the case of s = 0 are commonly used as a simple approximation for the electronic structureof a graphene layer:

Eg2D(kx, ky) = ±t

{

1 + 4 cos

(√3kxa

2

)

cos

(kya

2

)

+ 4 cos2(

kya

2

)}1/2

. (B.8)

The simple approximation given by Eq. (B.8) is used next to obtain a simple approxima-tion for the electronic dispersion relations for carbon nanotubes, and provides an excellentfirst approximation for the analysis of presently available experiments on carbon nanotubes.

B.2 Single Wall Carbon Nanotubes

In §B.2 we briefly review the structure of single wall carbon nanotubes and relate this struc-ture to the 2D graphene sheet discussed in §B.1, while §B.2.1 gives the electronic structureof the single wall carbon nanotube, as obtained from the tight binding approximation andfrom E(k) for the graphene sheet, given by Eq.B.8.

B.2.1 Structure

A single-wall carbon nanotube can be described as a graphene sheet rolled into a cylindricalshape so that the structure is one-dimensional with axial symmetry, and in general exhibitsa spiral conformation, called chirality. The chirality, as defined in this appendix, is givenby a single vector called the chiral vector. To specify the structure of carbon nanotubes,we define several important vectors, which are derived from the chiral vector.

Chiral Vector: Ch

The structure of a single-wall carbon nanotube (see Fig. B.3) is specified by the vector

(−→OA in Fig. B.4) which corresponds to a section of the nanotube perpendicular to the

nanotube axis (hereafter we call this section the equator of the nanotube). In Fig. B.4, the

unrolled honeycomb lattice of the nanotube is shown, in which−→OB is the direction of the

nanotube axis, and the direction of−→OA corresponds to the equator. By considering the

crystallographically equivalent sites O, A, B, and B ′, and by rolling the honeycomb sheetso that points O and A coincide (and points B and B ′ coincide), a paper model of a carbon

nanotube can be constructed. The vectors−→OA and

−→OB define the chiral vector Ch and the

translational vector T of a carbon nanotube, respectively, as further explained below.The chiral vector Ch can be expressed by the real space unit vectors a1 and a2 (see

Fig. B.4) of the hexagonal lattice defined in Eq. (B.1):

Ch = na1 + ma2 ≡ (n, m), (n, m are integers, 0 ≤ |m| ≤ n). (B.9)

The specific chiral vectors Ch shown in Fig. B.3 are, respectively, (a) (5, 5), (b) (9, 0) and (c)(10, 5), and the chiral vector shown in Fig. B.4 is (4, 2). An armchair nanotube correspondsto the case of n = m, that is Ch = (n, n) [see Fig. B.3(a)], and a zigzag nanotube corresponds

192

(a)

(b)

(c)

Figure B.3: Classification of carbon nanotubes: (a) armchair, (b) zigzag, and (c) chiralnanotubes, showingcross-sections and caps for the 3 basic kinds of nanotubes.

a1

a2

O

A

B

B

T

Ch

θR

y

x

Figure B.4: The unrolled honeycomb lattice of a nanotube, showing the unit vectors ~a1 and~a2 for the graphene sheet. When we connect sites O and A, and B and B ′, a nanotube can

be constructed.−→OA and

−→OB define the chiral vector Ch and the translational vector T of

the nanotube, respectively. The rectangle OAB ′B defines the unit cell for the nanotube.The figure corresponds to Ch = (4, 2), d = dR = 2, T = (4,−5), N = 28, R = (1,−1).

193

to the case of m = 0, or Ch = (n, 0) [see Fig. B.3(b)]. All other (n, m) chiral vectorscorrespond to chiral nanotubes [see Fig. B.3(c)]. Because of the hexagonal symmetry ofthe honeycomb lattice, we need to consider only 0 < |m| < n in Ch = (n, m) for chiralnanotubes.

The diameter of the carbon nanotube, dt, is given by L/π, in which L is the circumfer-ential length of the carbon nanotube:

dt = L/π, L = |Ch| =√

Ch · Ch = a√

n2 + m2 + nm. (B.10)

It is noted here that a1 and a2 are not orthogonal to each other and that the inner productsbetween a1 and a2 yield:

a1 · a1 = a2 · a2 = a2, a1 · a2 =a2

2, (B.11)

where the lattice constant a = 1.42 A ×√

3 = 2.46 A of the honeycomb lattice is given inEq. (B.1).

The chiral angle θ (see Fig. B.4) is defined as the angle between the vectors Ch anda1, with values of θ in the range 0 ≤ |θ| ≤ 30◦, because of the hexagonal symmetry of thehoneycomb lattice. The chiral angle θ denotes the tilt angle of the hexagons with respectto the direction of the nanotube axis, and the angle θ specifies the spiral symmetry. Thechiral angle θ is defined by taking the inner product of Ch and a1, to yield an expressionfor cos θ:

cos θ =Ch · a1

|Ch||a1|=

2n + m

2√

n2 + m2 + nm, (B.12)

thus relating θ to the integers (n, m) defined in Eq. (B.9). In particular, zigzag and armchairnanotubes correspond to θ = 0◦ and θ = 30◦, respectively.

B.2.2 Translational Vector: T

The translation vector T is defined to be the unit vector of a 1D carbon nanotube. Thevector T is parallel to the nanotube axis and is normal to the chiral vector Ch in the

unrolled honeycomb lattice in Fig. B.4. The lattice vector T shown as−→OB in Fig. B.4 can

be expressed in terms of the basis vectors a1 and a2 as:

T = t1a1 + t2a2 ≡ (t1, t2), (where t1, t2 are integers). (B.13)

The translation vector T corresponds to the first lattice point of the 2D graphene sheet

through which the vector−→OB (normal to the chiral vector Ch) passes. From this fact, it is

clear that t1 and t2 do not have a common divisor except for unity. Using Ch · T = 0 andEqs. (B.9), (B.11), and (B.13), we obtain expressions for t1 and t2 given by:

t1 =2m + n

dR, t2 = −2n + m

dR(B.14)

where dR is the greatest common divisor (gcd) of (2m+n) and (2n+m). Also, by introducingd as the greatest common divisor of n and m, then dR can be related to d by2

dR =

{

d if n − m is not a multiple of 3d3d if n − m is a multiple of 3d.

(B.15)

2This relation is obtained by repeated use of the fact that when two integers, α and β (α > β), have acommon divisor, γ, then γ is also the common divisor of (α− β) and β (Euclid’s law). When we denote the

194

The length of the translation vector, T , is given by:

T = |T| =√

3L/dR, (B.16)

where the circumferential nanotube length L is given by Eq. (F.18). We note that the lengthT is greatly reduced when (n, m) have a common divisor or when (n − m) is a multiple of3d. In fact, for the Ch = (5, 5) armchair nanotube, we have dR = 3d = 15, T = (1,−1)[Fig. B.3(a)], while for the Ch = (9, 0) zigzag nanotube we have dR = d = 9, and T = (1,−2)[Fig. B.3(b)].

The unit cell of the 1D carbon nanotube is the rectangle OAB ′B defined by the vectorsCh and T (see Fig. B.4), while the unit vectors a1 and a2 define the area of the unit cellof 2D graphite. When the area of the nanotube unit cell |Ch × T| (where the symbol ×denotes the vector product operator) is divided by the area of a hexagon (|a1 × a2|), thenumber of hexagons per unit cell N is obtained as a function of n and m in Eq. (B.9) as:

N =|Ch × T||a1 × a2|

=2(m2 + n2 + nm)

dR=

2L2

a2dR, (B.17)

where L and dR are given by Eqs. (F.18) and (B.15), respectively, and we note that eachhexagon contains two carbon atoms. Thus there are 2N carbon atoms (or 2pz orbitals) ineach unit cell of the carbon nanotube.

Unit Cells and Brillouin Zones

The unit cell for a carbon nanotube in real space is given by the rectangle generated by thechiral vector Ch and the translational vector T, as is shown in OAB ′B in Fig. B.4. Sincethere are 2N carbon atoms in this unit cell, we will have N pairs of bonding π and anti-bonding π∗ electronic energy bands. Similarly the phonon dispersion relations will consistof 6N branches resulting from a vector displacement of each carbon atom in the unit cell.

Expressions for the reciprocal lattice vectors K2 along the nanotube axis and K1 in thecircumferential direction3 are obtained from the relation Ri ·Kj = 2πδij , where Ri and Kj

are, respectively, the lattice vectors in real and reciprocal space. Then, using Eqs. (B.14),(B.17), and the relations

Ch · K1 = 2π, T · K1 = 0,Ch · K2 = 0, T · K2 = 2π,

(B.18)

we get expressions for K1 and K2:

K1 =1

N(−t2b1 + t1b2), K2 =

1

N(mb1 − nb2), (B.19)

where b1 and b2 are the reciprocal lattice vectors of two-dimensional graphite given byEq. (B.2). In Fig. B.5, we show the reciprocal lattice vectors, K1 and K2, for a Ch =

greatest common divisor as γ = gcd(α, β), we get

dR = gcd(2m + n, 2n + m) = gcd(2m + n, n − m) = gcd(3m, n − m) = gcd(3d, n − m),

which gives Eq. (B.15).3Since nanotubes are one-dimensional materials, only K2 is a reciprocal lattice vector. K1 gives discrete

k values in the direction of Ch.

195

b1

b2

WK2

1KΓ M

K

WK

Figure B.5: The Brillouin zone of a carbon nanotube is represented by the line segment WW ′

which is parallel to K2. The vectors K1 and K2 are reciprocal lattice vectors correspondingto Ch and T, respectively. The figure corresponds to Ch = (4, 2), T = (4,−5), N = 28,K1 = (5b1 + 4b2)/28, K2 = (4b1 − 2b2)/28 (see text).

(4, 2) chiral nanotube. The first Brillouin zone of this one-dimensional material is the linesegment WW ′. Since NK1 = −t2b1+t1b2 corresponds to a reciprocal lattice vector of two-dimensional graphite, two wave vectors which differ by NK1 are equivalent. Since t1 and t2do not have a common divisor except for unity (see Sect. B.2.2), none of the N − 1 vectorsµK1 (where µ = 1, · · · , N − 1) are reciprocal lattice vectors of two-dimensional graphite.Thus the N wave vectors µK1 (µ = 0, · · · , N − 1) give rise to N discrete k vectors, asindicated by the N = 28 parallel line segments in Fig. B.5, which arise from the quantizedwave vectors associated with the periodic boundary conditions on Ch. The length of all theparallel lines in Fig. B.5 is 2π/T which is the length of the one-dimensional first Brillouinzone. For the N discrete values of the k vectors, N one-dimensional energy bands willappear. Because of the translational symmetry of T, we have continuous wave vectors inthe direction of K2 for a carbon nanotube of infinite length. However, for a nanotube offinite length Lt, the spacing between wave vectors is 2π/Lt.

B.3 Electronic Structure of Single-Wall Nanotubes

B.3.1 Zone-Folding of Energy Dispersion Relations

The electronic structure of a single-wall nanotube can be obtained simply from that oftwo-dimensional graphite. By using periodic boundary conditions in the circumferentialdirection denoted by the chiral vector Ch, the wave vector associated with the Ch directionbecomes quantized, while the wave vector associated with the direction of the translationalvector T (or along the nanotube axis) remains continuous for a nanotube of infinite length.Thus the energy bands consist of a set of one-dimensional energy dispersion relations whichare cross sections of those for two-dimensional graphite (see Fig. B.2).

When the energy dispersion relations of two-dimensional graphite, Eg2D(k) [see Eqs. (B.6)and/or (B.8)] at line segments shifted from WW ′ by µK1 (µ = 0, · · · , N − 1) are folded sothat the wave vectors parallel to K2 coincide with WW ′ as shown in Fig. B.5, N pairs of

196

Γ

Y

K1

K2

kx

kyK

Μ

W K

Μ

W

K

Figure B.6: The condition for metallic energy bands: if the ratio of the length of the vector−→Y K to that of K1 is an integer, metallic energy bands are obtained.

1D energy dispersion relations Eµ(k) are obtained, where N is given by Eq. (B.17). These1D energy dispersion relations are given by

Eµ(k) = Eg2D

(

kK2

|K2|+ µK1

)

, (µ = 0, · · · , N − 1, and − π

T< k <

π

T), (B.20)

corresponding to the energy dispersion relations of a single-wall carbon nanotube. The Npairs of energy dispersion curves given by Eq. (B.20) correspond to the cross sections of thetwo-dimensional energy dispersion surface shown in Fig. B.2, where cuts are made on thelines of kK2/|K2|+µK1. If for a particular (n, m) nanotube, the cutting line passes througha K point of the 2D Brillouin zone (Fig. B.1), where the π and π∗ energy bands of two-dimensional graphite are degenerate by symmetry, the one-dimensional energy bands havea zero energy gap. In this case, the density of states at the Fermi level has a finite value forthese carbon nanotubes, and they therefore are metallic. If, however, the cutting line doesnot pass through a K point, then the carbon nanotube is expected to show semiconductingbehavior, with a finite energy gap between the valence and conduction bands.

The condition for obtaining a metallic energy band is that the ratio of the length of the

vector−→Y K to that of K1 in Fig. B.6 is an integer.4 Since the vector

−→Y K is given by

−→Y K=

2n + m

3K1, (B.21)

4There are two inequivalent K and K ′ points in the Brillouin zone of 2D graphite as is shown in Fig. B.6and thus the metallic condition can also be obtained in terms of K ′. However, the results in that case areidentical to the case specified by Y K.

197

(6,0) (7,0) (8,0) (9,0)

(4,1) (5,1) (6,1) (7,1) (8,1) (9,1)

(3,2) (4,2) (5,2) (6,2) (7,2) (8,2) (9,2)

(4,3) (5,3) (6,3) (7,3) (8,3)

(4,4) (5,4) (6,4) (7,4) (8,4)

(5,5) (6,5) (7,5)

(6,6) (7,6)

(1,0)(0,0) (2,0) (10,0)(3,0) (4,0) (5,0)

(2,1)

(3,3)

(2,2)

(1,1) (3,1)

: metal : semiconductor

armchair

zigzag

Figure B.7: The carbon nanotubes (n, m) that are metallic and semiconducting, respec-tively, are denoted by open and solid circles on the map of chiral vectors (n, m). For verysmall diameter nanotubes (e.g., dt < 0.7 nm), the tight binding approximation is not suf-ficiently accurate, and more detailed approaches are needed. For example, small diameternanotubes, such as the (4,2) nanotube is predicted to be semiconducting by tight bindingapproximation, though more detailed calculations show (4,2) to be metallic and experimentsindicate that it may be superconducting.

the condition for metallic nanotubes is that (2n + m) or equivalently (n − m) is a multipleof 3.5 In particular, the armchair nanotubes denoted by (n, n) are always metallic, and thezigzag nanotubes (n, 0) are only metallic when n is a multiple of 3.

In Fig. B.7, we show which carbon nanotubes are metallic and which are semiconducting,denoted by open and solid circles, respectively. From Fig. B.7, it follows that approximatelyone third of the carbon nanotubes are metallic and the other two thirds are semiconducting.

B.3.2 Energy Dispersion of Armchair and Zigzag Nanotubes

To obtain explicit expressions for the dispersion relations, the simplest cases to consider arethe nanotubes having the highest symmetry, i.e. the achiral armchair and zigzag nanotubes.The appropriate periodic boundary conditions used to obtain the energy eigenvalues forthe (n, n) armchair nanotube define the small number of allowed wave vectors kx,q in thecircumferential direction

n√

3kx,qa = 2πq, (q = 1, . . . , 2n). (B.22)

5Since 3n is a multiple of 3, the remainders of (2n + m)/3 and (n − m)/3 are identical.

198

X Γk

-3

-2

-1

0

1

2

3

E(k

)/t

X Γk

-3

-2

-1

0

1

2

3

E(k

)/t

X Γk

-3

-2

-1

0

1

2

3

E(k

)/t

(c)(b)(a) a1u+e1u+

e2u+

e1u+

e1u-a1u-

a1g-e1g-

e2g-

e2g+e1g+a1g+

a1u+e1u+e2u+

e3u+

e4u+

a1u-e1u-e4u-e2u-e3u-,e3g-e2g-e4g-e1g-a1g-e4g+

e3g+

e2g+e1g+a1g+

a1u+e1u+e2u+

e3u+

e4u+

a2u+,a2u-,a1u-e1u-e2u-e4u-e3u-e3g-e4g-e2g-e1g-a2g-,a2g-,a1g-

e4g+

e3g+

e2g+e1g+a1g+

Figure B.8: One-dimensional energy dispersion relations for (a) armchair (5, 5), (b) zigzag(9, 0), and (c) zigzag (10, 0) carbon nanotubes labeled by the irreducible representations ofthe point group Dnd or Dnh (which describe the symmetry of these nanotubes), dependingon whether there are even or odd numbers of bands n at the Γ point (k = 0). The a-bands are nondegenerate and the e-bands are doubly degenerate at a general k-point. TheX points for armchair and zigzag nanotubes correspond to k = ±π/a and k = ±π/

√3a,

respectively. (See Eqs. B.23–B.25.)

Substitution of the discrete allowed values for kx,q given by Eq. (B.22) into Eq. (B.8) yieldsthe energy dispersion relations Ea

q (k) for the armchair nanotube, Ch = (n, n),

Eaq (k) = ±t

{

1 ± 4 cos

(qπ

n

)

cos

(ka

2

)

+ 4 cos2(

ka

2

)}1/2

,

(−π < ka < π), (q = 1, . . . , 2n)

(B.23)

in which the superscript a refers to armchair and k is a one-dimensional vector in thedirection of the vector K2 = (b1−b2)/2. This direction corresponds to the vector from theΓ point to the K point in the two-dimensional Brillouin zone of graphite6 [see Fig. B.1(b)].The resulting calculated 1D dispersion relations Ea

q (k) for the (5, 5) armchair nanotube areshown in Fig. B.8(a), where we see six dispersion relations for the conduction bands7 andan equal number for the valence bands.

Because of the degeneracy point between the valence and conduction bands at the bandcrossing which occurs at the Fermi energy, the (5, 5) armchair nanotube is thus a zero-gapsemiconductor which will exhibit metallic conduction at finite temperatures, because onlyinfinitesimal excitations are needed to excite carriers into the conduction band. All (n, n)armchair nanotubes have a band degeneracy between the highest valence band and thelowest conduction band at k = ±2π/(3a), where the bands cross the Fermi level. Thus, allarmchair nanotubes are expected to exhibit metallic conduction, similar to the behavior of2D graphene sheets.

6Note that K2 vector is not a reciprocal lattice vector of the 2D graphite.7The Fermi energy EF corresponds to E/t = 0. The upper half of Fig. B.8 corresponds to the unoccupied

conduction bands.

199

-1.0 -0.5 0.0 0.5 1.0kT/π

-1.0

-0.5

0.0

0.5

1.0

E(k

)/t

Ch=(9,6)

Figure B.9: Plot of the energy bands E(k) for the metallic 1D nanotube (n, m) = (9, 6) forvalues of the energy between −t and t, in dimensionless units E(k)/|t|. The Fermi level isat E = 0. The largest common divisor of (9,6) is d = 3, and the value of dR is dR = 3. Thegeneral behavior of the four energy bands intersecting at k = 0 is typical of the case wheredR = d.

The energy bands for the Ch = (n, 0) zigzag nanotube Ezq (k) can be obtained likewise

from Eq. (B.8) by writing the periodic boundary condition on ky as:

nky,qa = 2πq, (q = 1, . . . , 2n), (B.24)

to yield the 1D dispersion relations for the 4n states for the (n, 0) zigzag nanotube (denotedby the superscript z)

Ezq (k) = ±t

{

1 ± 4 cos

(√3ka

2

)

cos

(qπ

n

)

+ 4 cos2(

qπ

n

)}1/2

,(

− π√3

< ka <π√3

)

, (q = 1, . . . , 2n).

(B.25)

The resulting calculated 1D dispersion relations Ezq (k) for the (9, 0) and (10, 0) zigzag

nanotubes are shown in Figs. B.8(b) and (c), respectively. There is no energy gap for themetallic (9, 0) nanotube at k = 0, whereas the (10, 0) nanotube indeed shows an energygap. For a general (n, 0) zigzag nanotube, when n is a multiple of 3, the energy gap atk = 0 becomes zero; however, when n is not a multiple of 3, an energy gap opens at k = 0,as seen in Fig. B.8(c).

B.3.3 Dispersion of Chiral Nanotubes

Chiral nanotubes have usually much larger unit cells and, therefore a large number ofbranches in their dispersion relation. In Fig. B.9, we show dispersion relations for the (9, 6)

200

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Energy/γ0

0.0

0.5

1.0

DO

S [s

tate

s/un

it ce

ll of

gra

phite

]

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Energy/γ0

0.0

0.5

1.0

DO

S [s

tate

s/un

it ce

ll of

gra

phite

]

(a) (n,m)=(10,0)

(b) (n,m)=(9,0)

Figure B.10: Electronic 1D density of states per unit cell of a 2D graphene sheet for two(n, 0) zigzag nanotubes: (a) the (9, 0) nanotube which has metallic behavior, (b) the (10, 0)nanotube which has semiconducting behavior. Also shown as a dashed line in the figure isthe density of states for the 2D graphene sheet.

chiral nanotube. Since n − m is a multiple of 3, this chiral nanotube is metallic.

B.4 Density of States, Energy Gap

Of particular interest has been the energy dependence of the nanotube density of states, asshown in Fig. B.10 which compares the density of states for metallic (9,0) and semiconduct-ing (10,0) zigzag nanotubes. In this figure, we see that the density of states near the Fermilevel EF (located at E = 0) is different for metallic and semiconducting nanotubes. Thedensity of states at EF has a value of zero for semiconducting nanotubes, but is non-zero(and small) for metallic nanotubes. Also of great interest are the singularities in the 1Ddensity of states, corresponding to extrema in the E(k) relations. The comparison betweenthe 1D density of states for the nanotubes and the 2D density of states for a graphene layer isincluded in the figure. Another important result, pertaining to semiconducting nanotubes,

201

0.4 0.9 1.4 1.9 2.4 2.9 dt [nm]

0.0

1.0

2.0

3.0

Eii(d

t) [

eV]

E11

E22

E11

E33

γο =2.90 eV

M

S

S

S Γ

K

K K

M

K

M

M M

K

MM

K

(a) (b)

Figure B.11: (a) Calculated energy separations Eii(dt) between van Hove singularities i inthe 1D electronic density of states of the conduction and valence bands for all (n, m) valuesvs nanotube diameter 0.4 < dt < 3.0 nm, using a value for the carbon-carbon energy overlapintegral of γ0 = 2.9 eV and a nearest neighbor carbon-carbon distance aC−C = 1.42 A.Semiconducting (S) and metallic (M) nanotubes are indicated by crosses and open circles,respectively. The index i in the interband transitions Eii denotes the transition betweenthe van Hove singularities, with i = 1 being closest to the Fermi level. (b) Plot of the2D equi-energy contours of graphite, showing trigonal warping effects in the contours, aswe move from the K point in the K − Γ or K − M directions. The equi-energy contoursare circles near the K point and near the center of the Brillouin zone. But near the zoneboundary, the contours are straight lines which connect the nearest M points.

202

shows that their energy gap depends upon the reciprocal nanotube diameter dt, accordingto the relation Eg = (|t|aC−C)/dt , independent of the chiral angle of the semiconductingnanotube, where aC−C = a/

√3.

It is significant that every (n, m) nanotube has a different and unique set of energieswhere the singularities in the 1D electronic density of states occur. Figure B.11(a) shows aplot of the energy differences Eii between singularities i in the conduction and valence bandsfor every possible nanotube as a function of nanotube diameter, showing the uniqueness ofthe energies of these singularities in the density of states. This uniqueness arises from thetrigonal warping effect. Figure B.11(b) shows that the constant energy surfaces around theorigin (Γ point where k = 0) and around the K and K ′ points in the 2D Brillouin zone arecircular only near the Γ, K, and K ′ high symmetry points. Away from these symmetrypoints, trigonal warping effects become important, giving rise to a different set of singular-ities in the density of states, depending on the nanotube diameter and chirality. We canmeasure the Eii singularities in the density of states at the single nanotube level by the Ra-man effect, which shows a strong resonance with an individual (n, m) carbon nanotube whenthe laser excitation energy is equal to one of these singularities. Therefore, the resonanceRaman effect can be used to identify the (n, m) values for individual carbon nanotubes.Because of the unique properties of these particular low dimensional systems, spectroscopycan be used to obtain structural information about individual carbon nanotubes.

203

Appendix C

Harmonic Oscillators, Phonons,

and Electron-Phonon Interaction

C.1 Harmonic Oscillators

In this section we review the solution of the harmonic oscillator problem in quantum me-chanics using raising and lowering operators. This is aimed at providing a quick review asbackground for the lecture on phonon scattering processes and other topics in this course.

The Hamiltonian for the harmonic oscillator in one-dimension is written as:

H =p2

2m+

1

2κx2. (C.1)

We know classically that the frequency of oscillation is given by ω =√

κ/m so that

H =p2

2m+

1

2mω2x2 (C.2)

Define the lowering and raising operators a and a† respectively by

a =p − imωx√

2hmω(C.3)

a† =p + imωx√

2hmω(C.4)

Since [p, x] = h/i, then [a, a†] =1 so that

H =1

2m

[

(p + iωmx)(p − iωmx) + mhω

]

(C.5)

= hω[a†a + 1/2]. (C.6)

Let N = a†a denote the number operator and its eigenstates |n〉 so that N |n〉 = n|n〉 wheren is any real number. However

〈n|N |n〉 = 〈n|a†a|n〉 = 〈y|y〉 = n ≥ 0 (C.7)

204

Figure C.1: Simple harmonic oscillator with single spring.

where |y〉 = a|n〉 and the absolute value square of the eigenvector cannot be negative. Hencen is a positive number or zero.

Na|n〉 = a†aa|n〉 = (aa† − 1)a|n〉 = (n − 1)a|n〉 (C.8)

Hence a|n〉 = c|n − 1〉 and 〈n|a†a|n〉 = |c|2. However from Eq.C.7 〈n|a†a|n〉 = n so thatc =

√n and a|n〉 =

√n|n − 1〉. Since the operator a lowers the quantum number of the

state by unity, a is called the annihilation operator. Therefore n also has to be an integer,so that the null state is eventually reached by applying operator a for a sufficient numberof times.

Na†|n〉 = a†aa†|n〉 = a†(1 + a†a)|n〉 = (n + 1)a†|n〉 (C.9)

Hence a†|n〉 =√

n + 1|n + 1〉 so that a† is called a raising operator or a creation operator.Finally,

H|n〉 = hω[N + 1/2]|n〉 = hω(n + 1/2)|n〉 (C.10)

so the eigenvalues become

E = hω(n + 1/2), n = 0, 1, 2, . . . . (C.11)

C.2 Phonons

In this section we relate the lattice vibrations to harmonic oscillators and identify thequanta of the lattice vibrations with phonons. Consider the 1-D model of atoms connectedby springs (see Fig. C.1). The Hamiltonian for this case is written as:

H =N∑

s=1

(p2

s

2m+

1

2κ(xs+1 − xs)

2)

(C.12)

This equation doesn’t look like a set of independent harmonic oscillators since xs and xs+1

are coupled. Let

xs=1/√N ∑

k Qkeiksa

ps=1/√N ∑

k Pkeiksa.

(C.13)

These Qk, Pk’s are called phonon coordinates. It can be verified that the commutationrelation for momentum and coordinate implies a commutation relation between Pk and Qk′

[ps, xs′ ] =h

iδss′ =⇒ [Pk, Qk′ ] =

h

iδkk′ . (C.14)

205

Figure C.2: Schematic for a one dimensional phonon model and the corresponding dispersionrelation.

The Hamiltonian in phonon coordinates is:

H =∑

k

(1

2mP †

kPk +1

2mω2

kQ†kQk

)

(C.15)

with the dispersion relation given by

ωk =√

2κ(1 − cos ka) (C.16)

This is all in Kittel ISSP, pp. 611-613. (see Fig. C.2) Again let

ak =iP †

k + mωkQk√2hmωk

, (C.17)

a†k =−iPk + mωkQ

†k√

2hmωk(C.18)

so that the Hamiltonian is written as:

H =∑

k

hωk(a†kak + 1/2) ⇒ E =

∑

k

(nk + 1/2)hωk (C.19)

The quantum of energy hωk is called a phonon. The state vector of a system of phonons iswritten as |n1, n2, . . . , nk, . . .〉, upon which the raising and lowering operator can act:

ak|n1, n2, . . . , nk, . . .〉 =√

nk |n1, n2, . . . , nk − 1, . . .〉 (C.20)

a†k|n1, n2, . . . , nk, . . .〉 =√

nk + 1 |n1, n2, . . . , nk + 1, . . .〉 (C.21)

From Eq.C.21 it follows that the probability of annihilating a phonon of mode k is theabsolute value squared of the diagonal matrix element or nk.

C.3 Electron-Phonon Interaction

The basic Hamiltonian for the electron-lattice system is

H =∑

k

p2k

2m+

1

2

′∑

kk′

e2

|~rk − ~rk′ | +∑

i

P 2i

2M+

1

2

′∑

ii′

Vion(~Ri − ~Ri′)+∑

k,i

Vel−ion(~rk − ~Ri) (C.22)

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where the first two terms constitute Helectron, the third and fourth terms are denoted byHion and the last term is Helectron−ion. The electron-ion interaction term can be separatedinto two parts: the interaction of electrons with ions in their equilibrium positions, and anadditional term due to lattice vibrations:

Hel−ion = H0el−ion + Hel−ph (C.23)

∑

k,i

Vel−ion(~rk − ~Ri) =∑

k,i

Vel−ion[~rk − (~R0i + ~si)] (C.24)

where ~R0i is the equilibrium lattice site position and ~si is the displacement of the atoms

from their equilibrium positions in a lattice vibration so that

H0el−ion =

∑

k,i

Vel−ion(~rk − ~R0i ) (C.25)

andHel−ph = −

∑

k,i

~si · ~∇Vel−ion(~rk − ~R0i ). (C.26)

In solving the Hamiltonian we use an adiabatic approximation, which solves the elec-tronic part of the Hamiltonian by

(Helectron + H0el−ion)ψ = Eelψ (C.27)

and seeks a solution of the total problem as

Ψ = ψ(~r1, ~r2, · · · ~R1, ~R2, · · ·)ϕ(~R1, ~R2, · · ·) (C.28)

such that HΨ = EΨ. Here Ψ is the wave function for the electron-lattice system. Pluggingthis into the Eq.C.22, we find

EΨ = HΨ = ψ(Hion + Eel)ϕ −∑

i

h2

2Mi

(

ϕ∇2i ψ + 2~∇iϕ · ~∇iψ

)

(C.29)

Neglecting the last term, which is small, we have

Hionϕ = (E − Eel)ϕ (C.30)

Hence we have decoupled the electron-lattice system.

(Helectron + H0el−ion)ψ = Eelψ (C.31)

which gives us the energy band structure and ψ satisfies Bloch’s theorem while ϕ is thewave function for the ions

Hionϕ = Eionϕ (C.32)

which gives us phonon spectra and harmonic oscillator like wave functions, as we havealready seen in §C.2.

The discussion has thus far left out the electron-phonon interaction Hel−ph

Hel−ph = −∑

k,i

~si · ~∇Vel−ion(~rk − ~R0i ) (C.33)

207

which is then treated as a perturbation. Since the displacement vector can be written interms of the normal coordinates Q~q,j

~si =1√NM

∑

~q,j

Q~q,jei~q·~R0

i ej (C.34)

where j denotes the polarization index, N is the total number of ions and M is the ionmass. Hence

Hel−ph = −∑

k,i

1√NM

∑

~q,j

Q~q,jei~q·~R0

i ej · ~∇Vel−ion(~rk − ~R0i ) (C.35)

where the normal coordinate can be expressed in terms of the lowering and raising operators

Q~q,j =

(h

2ω~q,j

) 1

2

(a~q,j + a†−~q,j). (C.36)

Writing out the time dependence explicitly,

a~q,j(t) = a~q,je−iω~q,jt (C.37)

a†~q,j(t) = a†~q,jeiω~q,jt (C.38)

we obtain

Hel−ph = −∑

~q,j

(h

2NMω~q,j

) 1

2

(a~q,je−iω~q,jt + a†~q,je

iω~q,jt)

×∑

k,i

(ei~q·~R0i + e−i~q·~R0

i )ej~∇Vel−ion(~rk − ~R0

i ) (C.39)

= −∑

~q,j

(h

2NMω~q,j

) 1

2

a~q,j

∑

k,i

ejei(~q ~R0

i −ω~q,jt) · ~∇Vel−ion(~rk − ~R0i )

+ c.c.) (C.40)

If we are only interested in the interaction between one electron and a phonon on a particularbranch, say the longitudinal acoustic (LA) branch, then we drop the summation over j andk

Hel−ph = −(

h

2NMω~q

) 1

2

(

a~q

∑

i

eei(~q·~R0i −ω~qt) · ~∇Vel−ion(~r − ~R0

i ) + c.c

)

(C.41)

where the first term in the bracket corresponds to the phonon absorption and the c.c. termcorresponds to the phonon emission.

With Hel−ph at hand, we can solve transport problems (e.g., τ due to phonon scattering)and optical problems (e.g., indirect transitions) exactly since all of these problems involvethe matrix element 〈f |Hel−ph|i〉 of Hel−ph linking states |i〉 and |j〉.

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Appendix D

Artificial Atoms

PHYSICS TODAY JANUARY 1993Marc A. KastnerMarc Kastner is the Donner Professor of Science in the department of physics at the Mas-sachusetts Institute of Technology, in Cambridge.

The charge and energy of a sufficiently small particle of metal or semiconductor arequantized just like those of an atom. The current through such a quantum dot or one-electron transistor reveals atom-like features in a spectacular way.

The wizardry of modern semiconductor technology makes it possible to fabricate par-ticles of metal or “pools” of electrons in a semiconductor that are only a few hundredangstroms in size. Electrons in these structures can display astounding behavior. Suchstructures, coupled to electrical leads through tunnel junctions, have been given vari-ous names: single electron transistors, quantum dots, zero-dimensional electron gases andCoulomb islands. In my own mind, however, I regard all of these as artificial atoms-atomswhose effective nuclear charge is controlled by metallic electrodes. Like natural atoms, thesesmall electronic systems contain a discrete number of electrons and have a discrete spectrumof energy levels. Artificial atoms, however, have a unique and spectacular property: Thecurrent through such an atom or the capacitance between its leads can vary by many ordersof magnitude when its charge is changed by a single electron. Why this is so, and how wecan use this property to measure the level spectrum of an artificial atom, is the subject ofthis article.

To understand artificial atoms it is helpful to know how to make them. One way toconfine electrons in a small region is by employing material boundaries by surrounding ametal particle with insulator, for example. Alternatively, one can use electric fields to confineelectrons to a small region within a semiconductor. Either method requires fabricating verysmall structures. This is accomplished by the techniques of electron and x-ray lithography.Instead of explaining in detail how artificial atoms are actually fabricated, I will describethe various types of atoms schematically.

Figures D.1a and D.1b show two kinds of what is sometimes called, for reasons thatwill soon become clear, a single-electron transistor. In the first type (figure D.1a), whichI call the all-metal artificial atom,1 electrons are confined to a metal particle with typicaldimensions of a few thousand angstroms or less. The particle is separated from the leadsby thin insulators, through which electrons must tunnel to get from one side to the other.The leads are labeled “source” and “drain” because the electrons enter through the former

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and leave through the latter the same way the leads are labeled for conventional field effecttransistors, such as those in the memory of your personal computer. The entire structuresits near a large, well-insulated metal electrode, called the gate.

Figure D.1b shows a structures2 that is conceptually similar to the all-metal atom butin which the confinement is accomplished with electric fields in gallium arsenide. Like theall-metal atom, it has a metal gate on the bottom with an insulator above it; in this type ofatom the insulator is AlGaAs. When a positive voltage Vg is applied to the gate, electronsaccumulate in the layer of GaAs above the AlGaAs. Because of the strong electric field at theAlGaAs-GaAs interface, the electrons’ energy for motion perpendicular to the interface isquantized, and at low temperatures the electrons move only in the two dimensions parallel tothe -interface. The special feature that makes this an artificial atom is the pair of electrodeson the top surface of the GaAs. When a negative voltage is applied between these andthe source or drain, the electrons are repelled and cannot accumulate underneath them.Consequently the electrons are confined in a narrow channel between the two electrodes.Constrictions sticking but into the channel repel the electrons and create potential barriersat either end of the channel. A plot of a potential similar to the one seen by the electronsis shown in the inset in figure D.1. For an electron to travel from the source to the drainit must tunnel through the barriers. The “pool” of electrons that accumulates between thetwo constrictions plays the same role that the small particle plays in the all-metal atom, andthe potential barriers from the constrictions play the role of the thin insulators. Becauseone can control the height of these barriers by varying the voltage on the electrodes, Icall this type of artificial atom the controlled-barrier atom. Controlled- barrier atoms inwhich the heights of the two potential barriers can be varied independently have also beenfabricated.2 (The constrictions in these devices are similar to those used for measurementsof quantized conductance in narrow channels as reported in PHYSICS TODAY, November1988, page 21.) In addition, there are structures that behave like controlled-barrier atomsbut in which the barriers are caused by charged impurities or grain boundaries.2,4

Figure D.1c shows another, much simpler type of artificial atom. The electrons in alayer of GaAs are sandwiched between two layers of insulating AlGaAs. One or both ofthese insulators acts as a tunnel barrier. If both barriers are thin, electrons can tunnelthrough them, and the structure is analogous to the single-electron transistor without thegate. Such structures, usually called quantum dots, have been studied extensively.5,6 Tocreate the structure, one starts with two-dimensional layers like those in figure D.1b. Thecylinder can be made by etching away unwanted regions of the layer structure, or a metalelectrode on the surface, like those in figure D.1b, can be used to repel electrons everywhereexcept in a small circular section of GaAs. Although a gate electrode can be added to thiskind of structure, most of the experiments have bee done without one, so I call this thetwo-probe atom.

D.1 Charge quantization

One way to learn about natural atoms is to measure the energy required to add or removeelectrons. This usually done by photoelectron spectroscopy. For example the minimumphoton energy needed to remove an electron is the ionization potential, and the maximumenergy (photons emitted when an atom captures an electron is the electron affinity. Tolearn about artificial atoms we also measure the energy needed to add or subtract electron.

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Figure D.1: The many forms of artificial atoms include the all-metal atom (a), thecontrolled-barrier atom (b) and the two-probe atom, or “quantum dot” (c). Areas shownin blue are metallic, white areas are insulating, and red areas are semiconducting. Thedimensions indicated are approximate. The inset shows a potential similar to the one in thecontrolled-barrier atom, plotted as a function of position at the semiconductor-insulatorinterface. The electrons must tunnel through potential barriers caused by the two con-strictions. For capacitance measurements with a two-probe atom, only the source barrieris made thin enough for tunneling, but for current measurements both source and drainbarriers are thin.

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However, we do it by measuring the current through the artificial atom.Figure D.2 shows the current through a controller barrier atom7 as a function of the volt-

age Vg between the gate and the atom. One obtains this plot by applying very small voltagebetween the source and drain, just large enough to measure the tunneling conductance be-tween them. The results are astounding. The conductance( displays sharp resonances thatare almost periodic in Vg By calculating the capacitance between the artificial atom andthe gate we can show2,8 that the period is the voltage necessary to add one electron to theconfined pool of electrons. That is why we sometimes call the controller barrier atom asingle-electron transistor: Whereas the transistors in your personal computer turn on onlyon( when many electrons are added to them, the artificial atom turns on and off again everytime a single electron added to it.

A simple theory, the Coulomb blockade model, explains the periodic conductance reson-ances.9 (See PHYSICS TODAY, May 1988, page 19.) This model is quantitatively correctfor the all-metal atom and qualitatively correct for the controlled-barrier atom.10 To under-stand the model, think about how an electron in the all-metal atom tunnels from one leadonto the metal particle and then onto the other lead. Suppose the particle is neutral tobegin with. To add a charge Q to the particle requires energy Q2/2C, where C is the totalcapacitance between the particle and the rest of the system; since you cannot add less thanone electron the flow of current requires a Coulomb energy e2/2C. This energy barrier iscalled the Coulomb blockade. A fancier way to say this is that charge quantization leads toan energy gap in the spectrum of states for tunneling: For an electron to tunnel onto theparticle, its energy must exceed the Fermi energy of the contact by e2/2C, and for a holeto tunnel, its energy must be below the Fermi energy by the same amount. Consequentlythe energy gap has width e2/C. If the temperature is low enough that kT < e2/2C, neitherelectrons nor holes can flow from one lead to the other.

The gap in the tunneling spectrum is the difference between the “ionization potential”and the “electron affinity” of the artificial atom. For a hydrogen atom the ionizationpotential is 13.6 eV, but the electron affinity, the binding energy of H−, is only 0.75 eV.This large difference arises from the strong repulsive interaction between the two electronsbound to the same proton. Just as for natural atoms like hydrogen, the difference betweenthe ionization potential and electron affinity for artificial atoms arises from the electron-electron interactions; the difference, however, is much smaller for artificial atoms becausethey are much bigger than natural ones.

By changing the gate voltage Vg one can alter the energy required to add charge to theparticle. Vg is applied between the gate and the source, but if the drain-source voltage isvery small, the source, drain and particle will all be at almost the same potential. With Vg

applied, the electrostatic energy of a charge Q on the particle is

E = QVg + Q2/2C (D.1)

For negative charge Q, the first term is the attractive interaction between Q and the pos-itively charged gate electrode, and the second term is the repulsive interaction among thebits of charge on the particle. Equation D.1 shows that the energy as a function of Q is aparabola with its minimum at Q = −CVg. For simplicity I have assumed that the gate isthe only electrode that contributes to C; in reality, there are other contributions.7

By varying Vg we can choose any value of Q0, the charge that would minimize theenergy in equation D.1 if charge were not quantized. However, because the real charge

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Figure D.2: Conductance of a controlled-barrier atom as a function of the voltage Vg on thegate at a temperature of 60 mK. At low Vg (solid blue curve) the shape of the resonanceis given by the thermal distribution of electrons in the source that are tunneling onto theatom, but at high Vg a thermally broadened Lorentzian (red curve) is a better descriptionthan the thermal distribution alone (dashed blue curve). (Adapted from ref. 7.)

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Figure D.3: Total energy (top) and tunneling energies (bottom) for an artificial atom. Asvoltage is increased the charge Q0 for which the energy is minimized changes from −Ne to−(N + 1/4)e. Only the points corresponding to discrete numbers of electrons on the atomare allowed (dots on upper curves). Lines in the lower diagram indicate energies needed forelectrons or holes to tunnel onto the atom. When Q0 = −(N + 1/2)e the gap in tunnelingenergies vanishes and current can flow.

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is quantized, only discrete values of the energy E are possible. (See figure D.3.) WhenQ0 = −Ne, an integral number N of electrons minimizes E, and the Coulomb interactionresults in the same energy difference e2/2C for increasing or decreasing N by 1. For allother values of Q0 except Q0 = −(N + 1/2)e there is a smaller, but nonzero, energy foreither adding or subtracting an electron. Under such circumstances no current can flowat low temperature. However, if Q0 = −(N + 1/2)e the state with Q = −Ne and thatwith Q = (N + 1)e are degenerate, and the charge fluctuates between the two values evenat zero temperature. Consequently the energy gap in the tunneling spectrum disappears,and current can flow. The peaks in conductance are therefore periodic, occurring wheneverCVg = Q0 = −(N + 1/2)e, spaced in gate voltage by e/C.

As shown in figure D.3, there is a gap in the tunneling spectrum for all values of Vg exceptthe charge-degeneracy points. The more closely spaced discrete levels shown outside this gapare due to excited states of the electrons present on the artificial atom and will be discussedmore in the next section. As Vg is increased continuously, the gap is pulled down relativeto the Fermi energy until a charge degeneracy point is reached. On moving through thispoint there is a discontinuous change in the tunneling spectrum: The gap collapses and thenreappears shifted up by e2/C. Simultaneously the charge on the artificial atom increasesby 1 and the process starts over again. A charge-degeneracy point and a conductance peakare reached every time the voltage is increased by e/C, the amount necessary to add oneelectron to the artificial atom. Increasing the gate voltage of an artificial atom is thereforeanalogous to moving through the periodic table for natural atoms by increasing the nuclearcharge.

The quantization of charge on a natural atom is something we take for granted. However,if atoms were larger, the energy needed to add or remove electrons would be smaller, andthe number of electrons on them would fluctuate except at very low temperature. Thequantization of charge is just one of the properties that artificial atoms have in commonwith natural ones.

D.2 Energy quantization

The Coulomb blockade model accounts for charge quantization but ignores the quantizationof energy resulting from the small size of the artificial atom. This confinement of theelectrons makes the energy spacing of levels in the atom relatively large at low energies. Ifone thinks of the atom as a box, at the lowest energies the level spacings are of the orderh2/ma2, where a is the size of the box. At higher energies the spacings decrease for athree-dimensional atom because of the large number of standing electron waves possible fora given energy. If there are many electrons in the atom, they fill up many levels, and thelevel spacing at the Fermi energy becomes small. The all-metal atom has so many electrons(about 107) that the level spectrum is effectively continuous. Because of this, many expertsdo not regard such devices as “atoms,” but I think it is helpful to think of them as beingatoms in the limit in which the number of electrons is large. In the controlled-barrier atom,however, there are only about 30–60 electrons, similar to the number in natural atomslike krypton through xenon. Two-probe atoms sometimes have only one or two electrons.(There are actually many more electrons that are tightly bound to the ion cores of thesemiconductor, but those are unimportant because they cannot move.) For most cases,therefore, the spectrum of energies for adding an extra electron to the atom is discrete, just

215

as it is for natural atoms. That is why a discrete set of levels is shown in figure D.3.

One can measure the energy level spectrum directly by observing the tunneling currentat fixed Vg as a function of the voltage Vds between drain and source. Suppose we adjustVg so that, for example, Q0 = −(N + 1/4)e and then begin to increase Vds. The Fermilevel in the source rises in proportion to Vds relative to the drain, so it also rises relativeto the energy levels of the artificial atom. (See the inset to figure D.4a.) Current beginsto flow when the Fermi energy of the source is raised just above the first quantized energylevel of the atom. As the Fermi energy is raised further, higher energy levels in the atomfall below it, and more current flows because there are additional channels for electrons touse for tunneling onto the artificial atom. We measure an energy level by measuring thevoltage at which the current increases or, equivalently, the voltage at which there is a peakin the derivative of the current, dI/dVds. (We need to correct for the increase in the energyof the atom with Vds, but this is a small effect.) Many beautiful tunneling spectra of thiskind have been measured5 for two-terminal atoms. Figure D.4a shows one for a controlledbarrier atom.7

Increasing the gate voltage lowers all the energy levels in the atom by eVg, so that theentire tunneling spectrum shifts with Vg, as sketched in figure D.3. One can observe thiseffect by plotting the values of Vds at which peaks appear in dI/dVds. (See figure D.4b.) AsVg increases you can see the gap in the tunneling spectrum shift lower and then disappearat the charge-degeneracy point, just as the Coulomb blockade model predicts. You can alsosee the discrete energy levels of the artificial atom. For the range of Vs shown in figure D.4the voltage is only large enough to add or remove one electron from the atom; the discretelevels above the gap are the excited states of the atom with one extra electron, and thosebelow the gap are the excited states of the atom with one electron missing (one hole). Atstill higher voltages (not shown in figure D.4) one observes levels for two extra electrons orholes and so forth. The charge-degeneracy points are the values of Vg for which one of theenergy levels of the artificial atom is degenerate with the Fermi energy in the leads whenVds = 0, because only then can the charge of the atom fluctuate.

In a natural atom one has little control over the spectrum of energies for adding orremoving electrons. There the electrons interact with the fixed potential of the nucleusand with each other, and these two kinds of interaction determine the spectrum. In anartificial atom, however, one can change this spectrum completely by altering the atom’sgeometry and composition. For the all-metal atom, which has a high density of electrons,the energy spacing between the discrete levels is so small that it can be ignored. The highdensity of electrons also results in a short screening length for external electric fields, soelectrons added to the atom reside on its surface. Because of this, the electron-electroninteraction is always e2/C (where C is the classical geometrical capacitance), independentof the number of electrons added. This is exactly the case for which the Coulomb blockademodel was invented, and it works well: The conductance peaks are perfectly periodic in thegate voltage. The difference between the “ionization potential” and the “electron affinity”is e2/C, independent of the number of electrons on the atom.

In the controlled-barrier atom, as you can see from figure D.4, the level spacing is oneor two tenths of the energy gap. The conductance peaks are not perfectly periodic in gatevoltage, and the difference between ionization potential and electron affinity has a quantummechanical contribution. I will discuss this contribution a little later in more detail.

In the two-probe atom the electron-electron interaction can be made very small, so that

216

Figure D.4: Discrete energy levels of an artificial atom can be detected by varying thedrain-source voltage. When a large enough Vds is applied, electrons overcome the energygap and tunnel from the source to the artificial atom. (See inset of a.) a: Every time a newdiscrete state is accessible the tunneling current increases, giving a peak in dI/dVds. TheCoulomb blockade gap is the region between about –0.5 mV and + 0.3 mV where thereare no peaks. b: Plotting the positions of these peaks at various gate voltages gives thelevel spectrum. Note how the levels and the gap move downward as Vg increases, just assketched in the lower part of figure D.3. (Adapted from ref. 7.)

217

one can in principle reach the limit opposite to that of the all-metal atom. One can findthe energy levels of a two-probe atom by measuring the capacitance between its two leadsas a function of the voltage between them.6 When no tunneling occurs, this capacitance isthe series combination of the source-atom and atom-drain capacitances. For capacitancemeasurements, two-probe atoms are made with the insulating layer between the drain andatom so thick that current cannot flow under any circumstances. Whenever the Fermi levelin the source lines up with one of the energy levels of the atom, however, electrons can tunnelfreely back and forth between the atom and the source. This causes the total capacitance toincrease, because the source-atom capacitor is effectively shorted by the tunneling current.The amazing thing about this experiment is that a peak occurs in the capacitance everytime a single electron is added to the atom. (See figure D.5a.) The voltages at which thepeaks occur give the energies for adding electrons to the atom, just as the voltages for peaksin dI/dVds do for the controlled-barrier atom or for a two probe atom in which both thesource-atom barrier and the atom-drain barrier are thin enough for tunneling. The firstpeak in figure D.5a corresponds to the one-electron artificial atom.

Figure D.5b shows how the energies for adding electrons to a two-probe atom vary witha magnetic field perpendicular to the GaAs layer. In an all-metal atom the levels would beequally spaced, by e2/C, and would be independent of magnetic field because the electron-electron interaction completely determines the energy. By contrast, the levels of the two-probe atom are irregularly spaced and depend on the magnetic field in a systematic way.For the two-probe atom the fixed potential determines the energies at zero field. The levelspacings are irregular because the potential is not highly symmetric and varies at randominside the atom because of charged impurities in the GaAs and AlGaAs. It is clear that theelectron-electron interactions that are the source of the Coulomb blockade are not alwaysso important in the two-probe atom as in the all-metal and controlled-barrier atoms. Theirrelative importance depends in detail on the geometry.5

D.3 Artificial atoms in a magnetic field

Level spectra for natural atoms can be calculated theoretically with great accuracy, and itwould be nice to be able to do the same for artificial atoms. No one has yet calculated. anentire spectrum, like that in figure D.4a. However, for a simple geometry we can now predictthe charge-degeneracy points, the values of Vg corresponding to conductance peaks like thosein figure D.2. From the earlier discussion it should be clear that in such a calculation onemust take into account the electron’s interactions with both the fixed potential and theother electrons.

The simplest way to do this is with an extension of the Coulomb blockade model.11−13

It is assumed, as before, that the contribution to the gap in the tunneling spectrum fromthe Coulomb interaction is e2/C no matter how many electrons are added to the atom. Toaccount for the discrete levels one pretends that once on the atom, each electron interactsindependently with the fixed potential. All one has to do is solve for the energy levels of asingle electron in the fixed potential that creates the artificial atom and then fill those levelsin accordance with the Pauli exclusion principle. Because the electron-electron interactionis assumed always to be e2/C, this is called the constant-interaction model.

Now think about what happens when one adds electrons to a controlled-barrier atomby increasing the gate voltage while keeping Vds just large enough so one can measure the

218

Figure D.5: Capacitance of a two-probe atom that has only one barrier thin enough to allowtunneling. a: The capacitance has a peak every time a single electron is added to the atom.The positions of the peaks give the energy spectrum of the atom. b: Peaks in capacitanceplotted versus applied magnetic field. The green line indicates the rate of change of theenergy expected when the magnetic field dominates. (Adapted from ref. 6.)

219

conductance. When there are N − 1 electrons on the atom the N − 1 lowest energy levelsare filled. The next conductance peak occurs when the gate voltage pulls the energy of theatom down enough that the Fermi level in the source and drain becomes degenerate withthe Nth level. Only when an energy level is degenerate with the Fermi energy can currentflow; this is the condition for a conductance peak. When Vg is increased further and thenext conductance peak is reached, there are N electrons on the atom, and the Fermi level isdegenerate with the (N +1)-th level. Therefore to get from one peak to the next the Fermienergy must be raised by e2/C +(EN+1 −EN ), where EN , is the energy of the Nth level ofthe atom. If the energy levels are closely spaced the Coulomb blockade result is recovered,but in general the level spacing contributes to the energy between successive conductancepeaks.

It turns out that we can test the results of this kind of calculation best if a magnetic fieldis applied perpendicular to the GaAs layer. For free electrons in two dimensions, applyingthe magnetic field results in the spectrum of Landau levels with energies (n+1/2)hωc wherethe cyclotron frequency is ωc = eB/m∗c, and m∗ is the effective mass of the electrons. Inthe controlled barrier atom and the two-probe atom, we expect levels that behave likeLandau levels at high fields, with energies that increase linearly in B. This behavior occursbecause when the field is large enough the cyclotron radius is much smaller than the size ofthe electrostatic potential well that confines the electrons, and the electrons act as if theywere free. Levels shifting proportionally to B, as expected, are seen experimentally. (Seefigure D.5b.)

To calculate the level spectrum we need to model the fixed potential, the analog of thepotential from the nucleus of a natural atom. The simplest choice is a harmonic oscillatorpotential, and this turns out to be a good approximation for the controlled-barrier atom.Figure D.6a shows the calculated level spectrum as a function of magnetic field for non-interacting electrons in a two dimensional harmonic oscillator potential. At low fields theenergy levels dance around wildly with magnetic field. This occurs because some stateshave large angular momentum and the resulting magnetic moment causes their energies toshift up or down strongly with magnetic field. As the field is increased, however, thingssettle down. For most of the field range shown there are four families of levels, two movingup, the other two down. At the highest fields there are only two families, corresponding tothe two possible spin states of the electron.

Suppose we measure, in an experiment like the one whose results are shown in figure D.2,the gate voltage at which a specific peak occurs as a function of magnetic field. This valueof Vg is the voltage at which the Nth energy level is degenerate with the Fermi energyin the source and drain. A shift in the energy of the level will cause a shift in the peakposition. The blue line in figure D.6a is the calculated energy of the 39th level (chosen fairlyarbitrarily for illustration purposes), so it gives the prediction of the constant-interactionmodel for the position of the 39th conductance peak. As the magnetic field increases, levelsmoving up in energy cross those moving down, but the number of electrons is fixed, soelectrons jump from upward-moving filled levels to downward-moving empty ones. Thepeak always follows the 39th level, so it moves up and down in gate voltage.

Figure D.6b shows a measurement14 of Vg for one conductance maximum, like one ofthose in figure D.2, as a function of B. The behavior is qualitatively similar to that predictedby the constant-interaction model: The peak mover up and down with increasing B, andthe frequency of level crossings changes at the field where only the last two families of levels

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remain. However, at high B the frequency is predicted to be much lower than what isobserved experimentally. While the constant-interaction model is in qualitative agreementwith experiment, it is not quantitatively correct.

To anyone who has studied atomic physics, the constant-interaction model seems quitecrude. Even the simplest models used to calculate energies of many electron atoms de-termine the charge density and potential self- consistently. One begins by calculating thecharge density that would result from noninteracting electrons in the fixed potential, andthen one calculates the effective potential an electron sees because of the fixed potentialand the potential resulting from this charge density. Then one calculates the charge densityagain. One does this repeatedly until the charge density and potential are self-consistent.The constant-interaction model fails because it is not self- consistent. FigureD.6c showsthe results of a self-consistent calculation for the controlled-barrier atom.14 It is in goodagreement with experiment-much better agreement than the constant-interaction modelgives.

D.4 Conductance line shapes

In atomic physics, the next step after predicting energy levels is to explore how an atominteracts with the electromagnetic field, because the absorption and emission of photonsteaches us the most about atoms. For artificial atoms, absorption and emission of electronsplays this role, so we had better understand how this process works. Think about whathappens when the gate voltage in the controlled-barrier atom is set at a conductance peak,and an electron is tunneling back and forth between the atom and the leads. Since theelectron spends only a finite time τ on the atom, the uncertainty principle tells us that theenergy level of the electron has a width h/τ . Furthermore, since the probability of findingthe electron on the atom decays as et/τ , the level will have a Lorentzian line shape.

This line shape can be measured from the transmission probability spectrum T (E) ofelectrons with energy E incident on the artificial atom from the source. The spectrum isgiven by

T (E) =Γ2

Γ2 + (E − EN )2(D.2)

where Γ is approximately h/τ and EN is the energy of the Nth level. The probability thatelectrons are transmitted from the source to the drain is approximately,proportional15 tothe conductance G. In fact, G ' (e2/h)T , where e2/h is the quantum of conductance.It is easy to show that one must have G < e2/h for each of the barriers separately toobserve conductance resonances. (An equivalent argument is used to show that electronsin a disordered conductor are localized for G < e2/h. See, for example, the article by BorisL. Al’tshuler and Patrick A. Lee in PHYSICS TODAY, December 1988, page 36.) Thiscondition is equivalent to requiring that the separation of the levels is greater than theirwidth Γ.

Like any spectroscopy, our electron spectroscopy of artificial atoms has a finite resolution.The resolution is determined by the energy spread of the electrons in the source, which aretrying to tunnel into the artificial atom. These electrons are distributed according to theFermi-Dirac function,

f(E) =1

exp[(E − EF )/kT ] + 1(D.3)

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where EF is the Fermi energy. The tunneling current is given by

I =

∫e

hT (E)[f(E) − f(E − eVds)]dE. (D.4)

Equation D.4 says that the net current is proportional to the probability f(E)T (E) thatthere is an electron in the source with energy E and that the electron can tunnel betweenthe source and drain minus the equivalent probability for electrons going from drain tosource. The best resolution is achieved by making Vds ¿ kT . Then [f(E)− f(E − eVds)] 'eVds(df/dE), and I is proportional to Vds, so the conductance is I/Vds.

Figure D.2 shows that equations D.2–D.4 describe the experiments well: At low Vg,where Γ is much less than kT , the shape of the conductance resonance is given by theresolution function df/dE. But at higher Vg one sees the Lorentzian tails of the naturalline shape quite clearly. The width Γ depends exponentially on the height and width of thepotential barrier, as is usual for tunneling. The height of the tunnel barrier decreases withVg, which is why the peaks become broader with increasing Vg. Just as we have controlover the level spacing in artificial atoms, we also can control the coupling to the leads andtherefore the level widths. It is clear why the present generation of artificial atoms showunusual behavior only at low temperatures: When kT becomes comparable to the energyseparation between resonances, the peaks overlap and the features disappear.

D.5 Applications

The behavior of artificial atoms is so unusual that it is natural to ask whether they willbe useful for applications to electronics. Some clever things can be done., Because of theelectron-electron interaction, only one electron at a time can pass through the atom. Withdevices like the “turnstile” device16,17 shown on the cover of this issue the two tunnelbarriers can be raised and lowered independently. Suppose the two barriers are raised andlowered sequentially at a radio or microwave frequency ν. Then, with a small source-drainvoltage applied, an electron will tunnel onto the atom when the source-atom barrier islow and off it when the atom-drain barrier is low. One electron will pass in each timeinterval ν−1, producing a current eV. Other applications, such as sensitive electrometers,can be imagined.9,18 However, the most interesting applications may involve devices inwhich several artificial atoms are coupled together to form artificial molecules16,17,19 or inwhich many are coupled to form artificial solids. Because the coupling between the artificialatoms can be controlled, new physics as well as new applications may emerge. The age ofartificial atoms has only just begun.

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Figure D.6: Effect of magnetic field on energy level spectrum and conductance peaks. a:Calculated level spectrum for noninteracting electrons in a harmonic oscillator electrostaticpotential as a function of magnetic field. The blue line is the prediction that the constantinteraction model gives for the gate voltage for the 39th conductance peak. b: Measuredposition of a conductance peak in a controlled-barrier atom as a function of field. c: Positionof the 39th conductance peak versus field, calculated self-consistently. The scale in c doesnot match that in b because parameters in the calculation were not precisely matched tothe experimental conditions. (Adapted from Ref. 14.)

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References

1. T. A. Fulton, G. J. Dolan, Phys. Rev. Lett. 59, 109 (1987).2. U. Meirav, M. A. Kastner, S. J. Wind, Phys. Rev. Lett. 65, 771 (1990). M. A. Kastner,Rev. Mod. Phys. 64, 849 (1992).3. L. P. Kouwenhoven, N. C. van der Vaart, A. T. Johnson, W. Kool, C. J. P. M. Harmans,J. G. Williamson, A. A. M. Staring, C. T. Foxon, Z. Phys. B 85, 367 (1991), and refs.therein.4. V. Chandrasekhar, Z. Ovadyahu, R. A. Webb, Phys. Rev. Lett. 67, 2862 (1991). R. J.Brown, M. Pepper, H. Ahmed, D. G. Hasko, D. A. Ritchie, J. E. F. Frost, D. C. Peacock,G. A. C. Jones, J. Phys.: Condensed Matter 2, 2105 (1990).5. B. Su, V. J. Goldman, J. E. Cunningham, Science 255, 313 (1992). M. A. Reed, J. N.Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, A. E. Wetsel, Phys. Rev. Lett. 60,535 (1988). M. Tewordt, L. Martin-Moreno, J. T. Nicholls, M. Pepper, M. J. Kelly, V. J.Law, D. A. Ritchie, J. E. F. Frost, G. A. C. Jones, Phys. Rev. B 45, 14407 (1992).6. R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, S. J. Pearton, K. Baldwin, K.W. West, Phys. Rev. Lett. 68, 3088 (1992).7. E. B. Foxman, P. L. McEuen, U. Meirav, N. S. Wingreen, Y. Meir, P. A. Belk, N. R.Belk, M. A. Kastner, S. J. Wind, “The Effects of Quantum Levels on Transport Through aCoulomb Island,” MIT preprint (July 1992). See also A. T. Johnson, L. P. Kouwenhoven,W. de Jong, N.C. van der Vaart, C. J. P. M. Harmans, C. T. Faxon, Phys. Rev. Lett. 69,1592 (1992).8. A. Kumar, Surf. Sci. 263, 335 (1992). A. Kumar, S. E. Laux, F. Stern, Appl. Phys.Lett. 54, 1270 (1989).9. D. V. Averin, K. K. Likharev, in Mesmcopic Phenomena in Solids, B. L. Al’tshuler, P.A. Lee, R. A. Webb, eds., Elsevier, Amsterdam (1991), p. 173.10. H. van Houton, C. W. J. Beenakker, Phys. Rev. Lett. 63,1893 (1989).11. D. V. Averin, A. N. Korotkov, Zh. Eksp. Teor. Fiz. 97, 1661 (1990) [Sov. Phys. JETP70, 937 (1990)].12. Y. Meir, N. S. Wingreen, P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991).13. C. J. Beenakker, Phys. Rev. B 44, 1646 (1991).14 P.L. McEuen, E. B. Foxman, J. Kinaret, U. Meirav, M. A. Kastner, N. S. Wingreen, S.J. Wind, Phys. Rev. B 45, 11419 (1992).15. R. Landauer, IBM J. Res. Dev. 1, 223 (1957).16. L. P. Kouwenhoven, A. T. Johnson, N. C. van der Vaart, W. Kool, C. J. P. M. Harmans,C. T. Foxon, Phys. Rev. Lett. 67, 1626 (1991).17. L. J. Geerligs, V. F. Anderegg, P A. M. Holweg, J. E. Mooij, H. Pothier, D. Esteve, C.Urbina, M. H. Devoret, Phys. Rev. Lett. 64, 2691 (1990).18. H. Grabert, M. H. Devoret, eds., Single Charge Tunneling, Plenum, New York (1992).19. R.J. Haug, J.M. Hong, K.Y.Lee, Surf. Sci. 263, 415 (1991).

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Appendix E

Transport in 1D materials

E.1 Carbon Nanotubes

The nanometer dimensions of the carbon nanotubes together with the unique electronicstructure of a graphene sheet make the electronic properties of these one-dimensional struc-tures highly unusual. This Chapter reviews some theoretical work on the relation betweenthe atomic structure and the electronic and transport properties of single-walled carbonnanotubes. In addition to the ideal tubes, results on the quantum conductance of nan-otube junctions and tubes with defects will be discussed. On-tube metal-semiconductor,semiconductor-semiconductor, and metal-metal junctions have been studied. Other defectssuch as substitutional impurities and pentagon-heptagon defect pairs on tube walls areshown to produce interesting effects on the conductance. The effects of static externalperturbations on the transport properties of metallic nanotubes and doped semiconductingnanotubes are examined, with the metallic tubes being much less affected by long-rangedisorder. The structure and properties of crossed nanotube junctions and ropes of nan-otubes have also been studied. The rich interplay between the structural and the electronicproperties of carbon nanotubes gives rise to new phenomena and the possibility of nanoscaledevice applications.

E.2 Introduction

Carbon nanotubes are tubular structures that are typically several nanometers in diameterand many microns in length. This fascinating new class of materials was first discoveredby S. Iijima [1] in the soot produced in the arc-discharge synthesis of fullerenes. Becauseof their nanometer dimensions, there are many interesting and often unexpected propertiesassociated with these structures, and hence there is the possibility of using them to studynew phenomena and employing them in applications [2, 3, 4]. In addition to the multi-walledtubes, single-walled nanotubes [5, 6, 7], and ropes of close-packed single-walled tubes havebeen synthesized. [8] Also, carbon nanotubes may be filled with foreign materials [9, 10] orcollapsed into flat, flexible nanoribbons [11]. Carbon nanotubes are highly unusual electricalconductors, the strongest known fibers, and excellent thermal conductors. Many potentiallyimportant applications have been explored, including the use of nanotubes as nanoprobetips [12], field emitters [13, 14], storage or filtering media [15], and nanoscale electronic

225

devices [16, 17, 18, 19, 20, 21, 22, 23, 24]. Further, it has been found that nanotubes mayalso be formed with other layered materials [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].In particular, BN, BC3, and other BxCyNz nanotubes have been theoretically predicted[25, 26, 27, 28, 29] and experimentally synthesized [30, 31, 32, 33, 34, 35].

Many different aspects of carbon nanotubes are treated in the various sections of thisVolume. In this Chapter, we focus on a review of some selected theoretical studies on theelectronic and transport properties of carbon nanotube structures, in particular, those ofjunctions, impurities, and other defects. Structures such as ropes of nanotubes and crossednanotubes are also discussed.

The organization of the Chapter is as follows. Section 2 contains an introduction tothe geometric and electronic structure of ideal single-walled carbon nanotubes. Section 3gives a discussion of the electronic and transport properties of various on-tube structures.Topics presented include on-tube junctions, impurities, and local defects. On-tube metal-semiconductor, semiconductor-semiconductor, and metal-metal junctions may be formedby introducing topological structural defects. These junctions have been shown to behavelike nanoscale device elements. Other defects such as substitutional impurities and Stone–Wales defects on tube walls also are shown to produce interesting effects on the conductance.Crossed nanotubes provide another means to obtain junction behavior. The crossed-tubejunctions, nanotube ropes, and effects of long-range disorder are the subjects of Section 4.Intertube interactions strongly modify the electronic properties of a rope. The effects oflong-range disorder on metallic nanotubes are quite different from those on doped semicon-ducting tubes. Finally, a summary and some conclusions are given in Section 5.

E.3 Geometric and Electronic Structure of Carbon Nanotubes

In this Section, we give an introduction to the structure and electronic properties of thesingle-walled carbon nanotubes (SWNTs). Shortly after the discovery of the carbon nan-otubes in the soot of fullerene synthesis, single-walled carbon nanotubes were synthesized inabundance using arc discharge methods with transition metal catalysts [5, 6, 7]. These tubeshave quite small and uniform diameter, on the order of one nanometer. Crystalline ropesof single-walled nanotubes with each rope containing tens to hundreds of tubes of similardiameter closely packed have also been synthesized using a laser vaporization method [8]and other techniques, such as arc-discharge and CVD techniques. These developments haveprovided ample amounts of sufficiently characterized samples for the study of the funda-mental properties of the SWNTs. As illustrated in Fig. 1, a single-walled carbon nanotubeis geometrically just a rolled up graphene strip. Its structure can be specified or indexed byits circumferential periodicity [37]. In this way, a SWNT’s geometry is completely specifiedby a pair of integers (n, m) denoting the relative position ~c = n~a1+m~a2 of the pair of atomson a graphene strip which, when rolled onto each other, form a tube.

Theoretical calculations [?, 39, 40, 41] have shown early on that the electronic propertiesof the carbon nanotubes are very sensitive to their geometric structure. Although grapheneis a zero-gap semiconductor, theory has predicted that the carbon nanotubes can be metalsor semiconductors with different size energy gaps, depending very sensitively on the diameterand helicity of the tubes, i.e., on the indices (n, m). As seen below, the intimate connectionbetween the electronic and geometric structure of the carbon nanotubes gives rise to many ofthe fascinating properties of various nanotube structures, in particular nanotube junctions.

226

Figure E.1: Geometric struc-ture of an (n, m) single-walled carbon nanotube

The physics behind this sensitivity of the electronic properties of carbon nanotubes totheir structure can be understood within a band-folding picture. It is due to the uniqueband structure of a graphene sheet, which has states crossing the Fermi level at only 2inequivalent points in k-space, and to the quantization of the electron wavevector alongthe circumferential direction. An isolated sheet of graphite is a zero-gap semiconductorwhose electronic structure near the Fermi energy is given by an occupied π band and anempty π∗ band. These two bands have linear dispersion and, as shown in Fig. 2, meet atthe Fermi level at the K point in the Brillouin zone. The Fermi surface of an ideal graphitesheet consists of the six corner K points. When forming a tube, owing to the periodicboundary conditions imposed in the circumferential direction, only a certain set of ~k statesof the planar graphite sheet is allowed. The allowed set of k’s, indicated by the lines inFig. 2, depends on the diameter and helicity of the tube. Whenever the allowed k’s includethe point K, the system is a metal with a nonzero density of states at the Fermi level,resulting in a one-dimensional metal with 2 linear dispersing bands. When the point K isnot included, the system is a semiconductor with different size energy gaps. It is importantto note that the states near the Fermi energy in both the metallic and the semiconductingtubes are all from states near the K point, and hence their transport and other propertiesare related to the properties of the states on the allowed lines. For example, the conductionband and valence bands of a semiconducting tube come from states along the line closestto the K point.

The general rules for the metallicity of the single-walled carbon nanotubes are as follows:

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K

Figure E.2: (Top) Tight-binding band structure of graphene (a single basal plane ofgraphite). (Bottom) Allowed ~k-vectors of the (7,1) and (8,0) tubes (solid lines) mappedonto the graphite Brillouin zone.

(n, n) tubes are metals; (n, m) tubes with n − m = 3j, where j is a nonzero integer, arevery tiny-gap semiconductors; and all others are large-gap semiconductors. Strictly withinthe band-folding scheme, the n − m = 3j tubes would all be metals, but because of tubecurvature effects, a tiny gap opens for the case that j is nonzero. Hence, carbon nanotubescome in three varieties: large-gap, tiny-gap, and zero gap. The (n, n) tubes, also knownas armchair tubes, are always metallic within the single-electron picture, independent ofcurvature because of their symmetry. As the tube radius R increases, the band gaps of thelarge-gap and tiny-gap varieties decreases with a 1/R and 1/R2 dependence, respectively.Thus, for most experimentally observed carbon nanotube sizes, the gap in the tiny-gap vari-ety which arises from curvature effects would be so small that, for most practical purposes,all the n − m = 3j tubes can be considered as metallic at room temperature. Thus, inFig. 2, a (7,1) tube would be metallic, whereas a (8,0) tube would be semiconducting.

This band-folding picture, which was first verified by tight-binding calculations [38, 39,40], is expected to be valid for larger diameter tubes. However, for a small radius tube,because of its curvature, strong rehybridization among the σ and π states can modify theelectronic structure. Experimentally, nanotubes with a radius as small as 3.5 A have beenproduced. Ab initio pseudopotential local density functional (LDA) calculations [41] indeedrevealed that sufficiently strong hybridization effects can occur in small radius nanotubeswhich significantly alter their electronic structure. Strongly modified low-lying conductionband states are introduced into the band gap of insulating tubes because of hybridization ofthe σ∗ and π∗ states. As a result, the energy gaps of some small radius tubes are decreasedby more than 50%. For example, the (6,0) tube which is predicted to be semiconductingin the band-folding scheme is shown to be metallic. For nanotubes with diameters greater

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Figure E.3: Strain energy/atom for carbon nanotubes from ab initio total energy calcula-tions [44]

than 1 nm, these rehybridization effects are unimportant. Strong σ-π rehybridization canalso be induced by bending a nanotube [42]

Energetically, ab initio total energy calculations have shown that carbon nanotubes arestable down to very small diameters. Figure 3 depicts the calculated strain energy peratom for different carbon nanotubes of various diameters [41]. The strain energy scalesnearly perfectly as d−2 where d is the tube diameter (solid curve in Fig. 3), as would bethe case for rolling a classical elastic sheet. Thus, for the structural energy of the carbonnanotubes, the elasticity picture holds down to a subnanometer scale. The elastic constantmay be determined from the total energy calculations. This result has been used to analyzecollapsed tubes [11] and other structural properties of nanotubes. Also shown in Fig. 3 isthe energy/atom for a (6,0) carbon strip. It has an energy which is well above that of a(6,0) tube because of the dangling bonds on the strip edges. Because in general the energyper atom of a strip scales as d−1, the calculation predicts that carbon nanotubes will bestable with respect to the formation of strips down to below 4 A in diameter, in agreementwith classical force-field calculations [43].

There have been many experimental studies on carbon nanotubes in an attempt tounderstand their electronic properties. The transport experiments [19, 20, 45, 46, 47] in-volved both two- and four-probe measurements on a number of different tubes, includingmultiwalled tubes, bundles of single-walled tubes, and individual single-walled tubes. Mea-surements showed that there are a variety of resistivity behaviors for the different tubes,consistent with the above theoretical picture of having both semiconducting and metal-lic tubes. In particular, at low temperature, individual metallic tubes or small ropes ofmetallic tubes act like quantum wires [19, 20]. That is, the conduction appears to oc-cur through well-separated discrete electron states that are quantum-mechanically coherentover distances exceeding many hundreds of nanometers. At sufficiently low temperature,the system behaves like an elongated quantum dot.

Figure 4 depicts the experimental set up for such a low temperature transport measure-ment on a single-walled nanotube rope from Ref. [20]. At a few degrees Kelvin, the low-biasconductance of the system is suppressed for voltages less than a few millivolts, and there

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Figure E.4: Experimental set-up for the electrical measurement of a single-walled nanotuberope, visible as the diagonal curved line [20]

are dramatic peaks in the conductance as a function of gate voltage that modulates thenumber of electrons in the rope. (See Fig. 5.) These results have been interpreted in termsof single-electron charging and resonant tunneling through the quantized energy levels of thenanotubes. The data are explained quite well using the band structure of the conductingelectrons of a metallic tube, but these electrons are confined to a small region defined eitherby the contacts or by the sample length, thus leading to the observed quantum confinementeffects of Coulomb blockade and resonant tunneling.

There have also been high resolution low temperature scanning tunneling microscopy(STM) studies, which directly probe the relationship between the structural and electronicproperties of the carbon nanotubes [48, 49]. Figure 6 is a STM image for a single carbonnanotube at 77 K on the surface of a rope. In these measurements, the resolution of themeasurements allowed for the identification of the individual carbon rings. From the orien-tation of the carbon rings and the diameter of the tube, the geometric structure of the tubedepicted in Fig. 6 was deduced to be that of a (11,2) tube. Measurement of the normalizedconductance in the scanning tunneling spectroscopy (STS) mode was then used to obtainthe local density of states (LDOS). Data on the (11,2) and the (12,3) nanotubes gave aconstant density of states at the Fermi level, showing that they are metals as predicted bytheory. On another sample, a (14,−3) tube was studied. Since 14+3 is not equal to 3 timesan integer, it ought to be a semiconductor. Indeed, the STS measurement gives a band gapof 0.75 eV, in very good agreement with calculations.

The electronic states of the carbon nanotubes, being band-folded states of graphene, leadto other interesting consequences, including a striking geometry dependence of the electricpolarizability. Figure 7 presents some results from a tight-binding calculation for the staticpolarizabilities of carbon nanotubes in a uniform applied electric field [50]. Results for 17single-walled tubes of varying size and chirality, and hence varying band gaps, are given.The unscreened polarizability α0 is calculated within the random phase approximation.The cylindrical symmetry of the tubes allows the polarizability tensor to be divided into

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Figure E.5: Measured conductance of a single-walled carbon nanotube rope as a functionof gate voltage [20]

Figure E.6: STM images at 77 K of a single-walled carbon nanotube at the surface of a rope[49]

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Figure E.7: Calculated static polarizability of single wall carbon nanotubes, showing resultsboth for α0⊥ vs R2 on the left and α‖ vs R/E2

g on the right [50]

components perpendicular to the tube axis, α0⊥, and a component parallel to the tube axis,α0‖. Values for α0⊥ predicted within this model are found to be totally independent of theband gap Eg and to scale linearly as R2, where R is the tube radius. The latter dependencemay be understood from classical arguments, but the former is rather unexpected. Theinsensitivity of α0⊥ to Eg results from selection rules in the dipole matrix elements betweenthe highest occupied and the lowest unoccupied states of these tubes. On the other hand,Fig. 7 shows that α0‖ is proportional to R/E2

g , which is consistent with the static dielectricresponse of standard insulators. Also, using arguments analogous to those for C60 [51, 52],local field effects relevant to the screened polarizability tensor α may be included classically,resulting in a saturation of α⊥ for large α0⊥, but leaving α‖ unaffected. Thus, in general,the polarizability tensor of a carbon nanotube is expected to be highly anisotropic withα‖ À α⊥. And the polarizability of small gap tubes is expected to be greatly enhancedamong tubes of similar radius.

Just as for the electronic states, the phonon states in carbon nanotubes are also quan-tized into phonon subbands. This has led to a number of interesting phenomena [2, 3] whichare discussed elsewhere in this Volume [53]. Here we mention several of them. It has beenshown that twisting motions of a tube can lead to the opening up of a minuscule gap at theFermi level, leading to the possibility of strong coupling between the electronic states andthe twisting modes or twistons [54]. The heat capacity of the nanotubes is also expected toshow a dimensionality dependence. Analysis [55] shows that the phonon contributions dom-inate the heat capacity, with single-walled carbon nanotubes having a Cph ∼ T dependenceat low temperature. The temperature below which this should be observable decreases withincreasing nanotube radius R, but the linear T dependence should be accessible to experi-mental investigations with presently available samples. In particular, a tube with a 100 Aradius should have Cph ∼ T for T < 7 K. Since bulk graphite has Cph ∼ T 2−3, a sampleof sufficiently small radius tubes should show a deviation from graphitic behavior. Multi-walled tubes, on the other hand, are expected to show a range of behavior intermediatebetween Cph ∼ T and Cph ∼ T 2−3, depending in detail on the tube radii and the numberof concentric walls.

In addition to their fascinating electronic properties, carbon nanotubes are found tohave exceptional mechanical properties [56]. Both theoretical [57, 58, 59, 60, 61, 62, 63] andexperimental [64, 65] studies have demonstrated that they are the strongest known fibers.Carbon nanotubes are expected to be extremely strong along their axes because of thestrength of the carbon-carbon bonds. Indeed, the Young’s modulus of carbon nanotubes

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has been predicted and measured to be more than an order of magnitude higher than that ofsteel and several times that of common commercial carbon fibers. Similarly, BN nanotubesare shown [66] to be the world’s strongest large-gap insulating fiber.

E.4 Electronic and Transport Properties of On-tube Struc-

tures

In this section, we discuss the electronic properties and quantum conductance of nanotubestructures that are more complex than infinitely long, perfect nanotubes. Many of thesesystems exhibit novel properties and some of them are potentially useful as nanoscale de-vices.

E.4.1 Nanotube junctions

Since carbon nanotubes are metals or semiconductors depending sensitively on their struc-tures, they can be used to form metal-semiconductor, semiconductor-semiconductor, ormetal-metal junctions. These junctions have great potential for applications since they areof nanoscale dimensions and made entirely of a single element. In constructing this kindof on-tube junction, the key is to join two half-tubes of different helicity seamlessly witheach other, without too much cost in energy or disruption in structure. It has been shownthat the introduction of pentagon-heptagon pair defects into the hexagonal network of asingle carbon nanotube can change the helicity of the carbon nanotube and fundamentallyalter its electronic structure [16, 17, 18, 67, 68, 69, 70, 71]. This led to the prediction thatthese defective nanotubes behave as the desired nanoscale metal-semiconductor Schottkybarriers, semiconductor heterojunctions, or metal-metal junctions with novel properties,and that they could be the building blocks of nanoscale electronic devices.

In the case of nanotubes, being one-dimensional structures, a local topological defectcan change the properties of the tube at an infinitely long distance away from the defect.In particular, the chirality or helicity of a carbon nanotube can be changed by creatingtopological defects into the hexagonal network. The defects, however, must induce zeronet curvature to prevent the tube from flaring or closing. The smallest topological defectwith minimal local curvature (hence less energy cost) and zero net curvature is a pentagon-heptagon pair [16, 17, 18, 67, 68, 69, 70, 71]. Such a pentagon-heptagon defect pair withits symmetry axis nonparallel to the tube axis changes the chirality of a (n, m) tube bytransferring one unit from n to m or vice versa. If the pentagon-heptagon defect pair isalong the (n, m) tube axis, then one unit is added or subtracted from m. Figure 8 depicts a(8,0) carbon tube joined to a (7,1) tube via a 5-7 defect pair. This system forms a quasi-1Dsemiconductor/metal junction, since within the band-folding picture the (7,1) half tube ismetallic and the (8,0) half tube is semiconducting.

Figures 9 and 10 show the calculated local density of states (LDOS) near the (8,0)/(7,1)junction. These results are from a tight-binding calculation for the π electrons [16]. Inboth figures, the bottom panel depicts the density of states of the perfect tube, with thesharp features corresponding to the van Hove singularities of a quasi-1D system. The otherpanels show the calculated LDOS at different distances away from the interface, with cell 1being the closest to the interface in the semiconductor or side and ring 1 the closest to theinterface in the metal side. Here, cell refers to one unit cell of the tube and “ring” refers

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Figure E.8: Atomic structure of an (8,0)/(7,1) carbon nanotube junction. The large light-gray balls denote the atoms forming the heptagon-pentagon pair [16]

to a ring of atoms around the circumference. These results illustrate the spatial behaviorof the density of states as it transforms from that of a metal to that of a semiconductoracross the junction. The LDOS very quickly changes from that of the metal to that ofthe semiconductor within a few rings of atoms as one goes from the metal side to thesemiconductor side. As the interface is approached, the sharp van Hove singularities of themetal are diluted. Immediately on the semiconductor side of the interface, a different setof singular features, corresponding to those of the semiconductor tube, emerges. There is,however, still a finite density of states in the otherwise bandgap region on the semiconductorside. These are metal induced gap states [72], which decay to zero in about a few A into thesemiconductor. Thus, the electronic structure of this junction is very similar to that of abulk metal-semiconductor junction, such as Al/Si, except it has a nanometer cross-sectionand is made out of entirely the element carbon.

Similarly, semiconductor-semiconductor and metal-metal junctions may be constructedwith the proper choices of tube diameters and pentagon-heptagon defect pairs. For example,by inserting a 5-7 pair defect, a (10,0) carbon nanotube can be matched to a (9,1) carbonnanotube [16]. Both of these tubes are semiconductors, but they have different bandgaps.The (10,0)/(9,1) junction thus has the electronic structure of a semiconductor heterojunc-tion. In this case, owing to the rather large structural distortion at the interface, thereare interesting localized interface states at the junction. Theoretical studies have also beencarried out for junctions of B-C-N nanotubes [73], showing very similar behaviors as thecarbon case, and for other geometric arrangements, such as carbon nanotube T-junctions,where one tube joins to the side of another tube perpendicularly to form a “T” structure[74].

Calculations have been carried out to study the quantum conductance of the carbonnanotube junctions. Typically these calculations are done within the Landauer formalism[75, 76]. In this approach, the conductance is given in terms of the transmission matrixof the propagating electron waves at a given energy. In particular, the conductance ofmetal-metal nanotube junctions is shown to exhibit a quite interesting new effect whichdoes not have an analog in bulk metal junctions [67]. It is found that certain configurationsof pentagon-heptagon pair defects in forming the junction completely stop the flow of elec-trons, while other arrangements permit the transmission of current through the junction.Such metal-metal junctions thus have the potential for use as nanoscale electrical switches.This phenomenon is seen in the calculated conductance of a (12,0)/(6,6) carbon nanotubejunction in Fig. 11. Both the (12,0) and (6,6) tubes are metallic within the tight-bindingmodel, and they can be matched perfectly to form a straight junction. However, the con-

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Figure E.9: Calculated LDOS of the (8,0)/(7,1) metal-semiconductor junction at the semi-conductor side. From top to bottom, LDOS at cells 1, 2, and 3 of the (8,0) side. Cell 1 isat the interface [16]

Figure E.10: Calculated LDOS of the (8,0)/(7,1) metal-semiconductor junction at the metalside. From top to bottom, the LDOS at rings 1, 2, and 3 of the (7,1) side. Ring 1 is at theinterface [16]

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Figure E.11: Calculated results for the (12,0)/(6,6) metal-metal junction. Top: conductanceof a matched tube (solid line), a perfect (12,0) tube (dashed line), and a perfect (6,6) tube(dotted line). Center: LDOS at the interface on the (12,0) side (full line ) and of the perfect(12,0) tube (dotted line). Bottom: LDOS at the interface on the (6,6) side (full line) andof the perfect (6,6) tube (dashed line) [67]

ductance is zero for electrons at the Fermi level, EF . This peculiar effect is not due to alack of density of states at EF . As shown in Fig. 11, there is finite density of states at EF

everywhere along the whole length of the total system for this junction. The absence ofconductance arises from the fact that there is discrete rotational symmetry along the axisof the combined tube. But, for electrons near EF , the states in one of the half tubes areof a different rotational symmetry from those in the other half tube. As an electron propa-gates from one side to the other, the electron encounters a symmetry gap and is completelyreflected at the junction.

The same phenomenon occurs in the calculated conductance of a (9,0)/(6,3) metal-metalcarbon nanotube junction. However, in forming this junction, there are two distinct ways tomatch the two halves, either symmetrically or asymmetrically. In the symmetric matchedgeometry, the conductance is zero at EF for the same symmetry reason as discussed above.(See Fig. 12.) But, in the asymmetric matched geometry, the discrete rotational symmetryof the total system is broken and the electrons no longer have to preserve their rotationalquantum number as they travel across the junction. The conductance for this case is nownonzero. Consequently, in some situations, bent junctions can conduct better than straightjunctions for the nanotubes. This leads to the possibility of using these metal-metal orother similar junctions as nanoswitches or strain gauges, i.e., one can imagine using somesymmetry breaking mechanisms such as electron-photon, electron-phonon or mechanicaldeformation to switch a junction from a non-conducting state to a conducting state [67].

Junctions of the kind discussed above may be formed during growth, but they can also

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Figure E.12: Calculated conductance of the (9,0)/(6,3) junction – matched system (solidline), perfect (6,3) tube (dotted line), and perfect (9,0) tube (dashed line) [67]

be generated by mechanical stress [77]. There is now considerable experimental evidenceof this kind of on-tube junction and device behavior predicted by theory. An experimentalsignature of a single pentagon-heptagon pair defect would be an abrupt bend between twostraight sections of a nanotube. Calculations indicate that a single pentagon-heptagon pairwould induce bend angles of roughly 0-15 degrees, with the exact value depending on theparticular tubes involved. Several experiments have reported sightings of localized bends ofthis magnitude for multiwalled carbon nanotubes [23, 78, 79]. Having several 5-7 defect pairsat a junction would allow the joining of tubes of different diameters and add complexityto the geometry. The first observation of nonlinear junction-like transport behavior wasmade on a rope of SWNTs [22], where the current-voltage properties were measured alonga rope of single-walled carbon nanotubes using a scanning tunneling microscopy tip andthe behavior shown in Fig. 13 was found in some samples. At one end of the tube, thesystem behaves like a semimetal showing a typical I-V curve of metallic tunneling, but aftersome distance at the other end it becomes a rectifier, presumably because a defect of theabove type has been introduced at some point on the tube. A more direct measurement wascarried out recently [23]. A kinked single-walled nanotube lying on several electrodes wasidentified and its electrical properties in the different segments were measured. The kinkwas indicative of two half tubes of different chiralities joined by a pentagon-heptagon defectpair. Figure 14 shows the measured I-V characteristics of a kinked nanotube. The inset isthe I-V curve for the upper segment showing that this part of the tube is a metal; but theI-V curve across the kink shows a rectifying behavior indicative of a metal-semiconductorjunction.

E.4.2 Impurities, Stone–Wales defects, and structural deformations in

metallic nanotubes

An unanswered question in the field has been why do the metallic carbon nanotubes havesuch long mean free paths. This has led to consideration of the effects of impurities anddefects on the conductance of the metallic nanotubes. We focus here on the (10,10) tubes;however, the basic physics is the same for all (n, n) tubes. In addition to tight-binding

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Figure E.13: Current-voltage characteristic mea-sured along a rope ofsingle-walled carbon nan-otubes. Panels A, B, C,and D correspond to suc-cessive different locationson the rope [22]

Figure E.14: Measured current-voltage characteristic of a kinked single-walled carbon nan-otube [23]

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studies, there are now first-principles calculations on the quantum conductance of nan-otube structures based on an ab initio pseudopotential density functional method with awavefunction matching technique [80, 81]. The advantages of the ab initio approach are thatone can obtain the self-consistent electronic and geometric structure in the presence of thedefects and, in addition to the conductance, obtain detailed information on the electronicwavefunction and current density distribution near the defect.

Several rather surprising results have been found concerning the effects of local defectson the quantum conductance of the (n, n) metallic carbon nanotubes [81]. For example,the maximum reduction in the conductance due to a local defect is itself often quantized,and this can be explained in terms of resonant backscattering by quasi-bound states of thedefect. Here we discuss results for three simple defects: boron and nitrogen substitutionalimpurities and the bond rotation or Stone–Wales defect. A Stone–Wales defect correspondsto the rotation of one of the bonds in the hexagonal network by 90 degrees, resulting in thecreation of a quite low energy double 5-7 defect pair, without changing the overall helicityof the tube.

Figure 15 depicts several results for a (10,10) carbon nanotube with a single boronsubstitutional impurity. The top panel is the calculated conductance as a function of theenergy of the electron. For a perfect tube, the conductance (indicated here by the dashedline) is 2 in units of the quantum of conductance, 2e2/h, since there are two conductancechannels available for the electrons near the Fermi energy. For the result with the boronimpurity, a striking feature is that the conductance is virtually unchanged at the Fermilevel of the neutral nanotube. That is, the impurity potential does not scatter incomingelectrons of this energy. On the other hand, there are two dips in the conductance below EF .The amount of the reduction at the upper dip is one quantum unit of conductance and itsshape is approximately Lorentzian. In fact, the overall structure of the conductance is welldescribed by the superposition of two Lorentzian dips, each with a depth of 1 conductancequantum. These two dips can be understood in terms of a reduction in conductance due toresonant backscattering from quasi-bound impurity states derived from the boron impurity.

The calculated results thus show that boron behaves like an acceptor with respect tothe first lower subband (i.e., the first subband with energy below the conduction states) andforms two impurity levels that are split off from the top of the first lower subband. Theseimpurity states become resonance states or quasi-bound states due to interaction with theconduction states. The impurity states can be clearly seen in the calculated LDOS nearthe boron impurity (middle panel of Fig. 15). The two extra peaks correspond to the twoquasi-bound states. The LDOS would be a constant for a perfect tube in the region betweenthe van Hove singularity of the first lower subband and that of the first upper subband.Because a (n, n) tube with a substitutional impurity still has a mirror plane perpendicularto the tube axis, the defect states have definite parity with respect to this plane. The upperenergy state (broader peak) in Fig. 15 has even parity and the lower energy state (narrowerpeak) has odd parity, corresponding to s-like and p-like impurity states, respectively.

The conductance behavior in Fig. 15 may be understood by examining how electronsin the two eigen-channels interact with the impurity. At the upper dip, an electron in oneof the two eigen-channels is reflected completely (99.9%) by the boron impurity, but anelectron in the other channel passes by the impurity with negligible reflection (0.1%). Thesame happens at the lower dip but with the behavior of the two eigen-channels switched.The bottom panel shows the calculated scattering phase shifts. The phase shift of the odd

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Figure E.15: Energy depen-dence of the calculated con-ductance, local density ofstates, and phase shifts ofa (10,10) carbon nanotubewith a substitutional boronimpurity [81]

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parity state changes rapidly as the energy sweeps past the lower quasi-bound state level,with its value passing through π/2 at the peak position of the quasi-bound state. The samechange occurs to the phase shift of the even parity state at the upper impurity-state energy.The total phase shift across a quasi-bound level is π in each case, in agreement with theFriedel sum rule. The picture is that an incoming electron with energy exactly in resonancewith the impurity state is being scattered back totally in one of the channels but not theother. This explains the exact reduction of one quantum of conductance at the dip. Theupper-energy impurity state has a large binding energy (over 0.1 eV) with respect to thefirst lower subband and hence is quite localized. It has an approximate extent of ∼10 A,whereas the lower impurity state has an extent of ∼250 A.

The results for a nitrogen substitutional impurity on the (10,10) tube are presented inFig. 16. Nitrogen has similar effects on the conductance as boron, but with opposite energystructures. Again, the conductance at the Fermi level is virtually unaffected, but there aretwo conductance dips above the Fermi level just below the first upper subband. Thus, thenitrogen impurity behaves like a donor with respect to the first upper subband, forming ans-like quasi-bound state with stronger binding energy and a p-like state with weaker bindingenergy. As in the case of boron, the reduction of one quantum unit of conductance at thedips is caused by the fact that, at resonance, the electron in one of the eigen channels isreflected almost completely by the nitrogen impurity but the electron in the other channelpasses by the impurity with negligible reflection. The LDOS near the nitrogen impurityshows two peaks corresponding to the two quasi-bound states. The phase shifts of the twoeigen channels show similar behavior as in the boron case.

For a (10,10) tube with a Stone–Wales or double 5-7 pair defect, the calculations also findthat the conductance is virtually unchanged for the states at the Fermi energy. Thus theseresults show that the transport properties of the neutral (n, n) metallic carbon nanotubeare very robust with respect to these kinds of intra-tube local defects. As in the impuritycase, there are two dips in the quantum conductance in the conduction band energy range,one above and one below the Fermi level. These are again due to the existence of defectlevels, and the reduction at the two dips is very close to one quantum of conductance forthe same reason, as discussed above. The symmetry of the Stone–Wales defect in this casedoes not cause mixing between the π and π∗ bands, and these two bands remain as eigenchannels in the defective system. The lower dip is due to a complete reflection of the π∗

band and the upper dip is due to complete reflection of the π band. This implies that theconductance of the nanotube, when there are more than one double 5-7 pair defect, wouldnot sensitively depend on their relative positions, but only on their total numbers, as longas the distance between defects is far enough to be able to neglect inter-defect interactions.The analysis of the phase shifts show that the lower quasi-bound state is even with respectto a mirror plane perpendicular to the tube axis, while the upper quasi-bound state is oddwith respect to the same plane.

The conductance of nanotubes can also be affected by structural deformations. Twotypes of deformations involving bending or twisting the nanotube structure have been con-sidered in the literature. It was found that a smooth bending of the nanotube does not leadto scattering [54], but formation of a local kink induces strong σ-π mixing and backscatter-ing similar to that discussed earlier for boron impurity [82]. Twisting has a much strongereffect [54]. A metallic armchair (n, n) nanotube upon twisting develops a band-gap whichscales linearly with the twisting angle up to the critical angle at which the tube collapses

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Figure E.16: Calculated con-ductance, local density ofstates and phase shifts ofa (10,10) carbon nanotubewith a substitutional nitro-gen impurity [81]

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Figure E.17: Energy depen-dence of the calculated con-ductance, local density ofstates, and phase shifts ofa (10,10) carbon nanotubewith a Stone–Wales defect[81]

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Figure E.18: Perspectiveview of a model of a rope of(10,10) carbon nanotubes

into a ribbon [82].

E.5 Nanotube Ropes, Crossed-Tube Junctions, and Effects

of Long-Range Perturbations

E.5.1 Ropes of nanotubes

Another interesting carbon nanotube system is that of ropes of single-walled carbon nan-otubes which have been synthesized in high yield [8]. These ropes, containing up to tensto hundreds of single-walled nanotubes in a close-packed triangular lattice, are made up oftubes of nearly uniform diameter, close to that of the (10,10) tubes (see Fig. 18.) Because ofthe rather weak interaction between these tubes, a naive picture would be that the packingof individual metallic nanotubes into ropes would not change their electronic propertiessignificantly. Theoretical studies [83, 84, 85] however showed that this is not the case fora rope of (10,10) carbon nanotubes. A broken symmetry of the (10,10) nanotube causedby interactions between tubes in a rope induces formation of a pseudogap in the densityof states of about 0.1 eV. The existence of this pseudogap alters many of the fundamentalelectronic properties of the rope.

As discussed above, an isolated (n, n) carbon nanotube has two linearly dispersing con-duction bands which cross at the Fermi level forming two “Dirac” points, as schematicallypresented in Fig. 19(a). This linear band dispersion in a one-dimensional system gives riseto a finite and constant density of electronic states at the Fermi energy. Thus, an (n, n)tube is a metal within the one-electron picture. The question of interest is: How doesthe electronic structure change when the metallic tubes are bundled up to form a closelypacked two-dimensional crystal, as in the case of the (10,10) ropes. In the calculation, alarge (10,10) rope is modeled by a triangular lattice of (10,10) tubes infinitely extended in

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Figure E.19: Band crossing and band repulsion. (a) Schematic band structure of an isolated(n, n) carbon nanotube near the Fermi energy. (b) Repulsion of bands due to the breakingof mirror symmetry.

the lateral directions. For such a system, the electronic states, instead of being contained inthe 1-D Brillouin zone of a single tube, are now extended to a three-dimensional irreducibleBrillouin zone wedge. If tube-tube interactions are negligibly small, the electronic energyband structure along any line in the wedge parallel to the rope axis would be exactly thesame as the band dispersion of an isolated tube. In particular, at the k-wavevector corre-sponding to the band crossing point, there will be a two-fold degenerate state at the Fermienergy. This allowed band crossing is due to the mirror symmetry of the (10,10) tube. Fora tube in a rope, this symmetry is however broken because of intertube interactions. Thebroken symmetry causes a quantum level repulsion and opens up a gap almost everywherein the Brillouin zone, as schematically shown in Fig. 19(b).

The band repulsion resulting from the broken-symmetry strongly modifies the densityof states (DOS) of the rope near the Fermi energy compared to that of an isolated (10,10)tube. The calculated DOS is presented in Fig. 20(a). Shown are the results for two cases:aligned and misaligned tubes in the rope. In both cases, there is a pseudogap of the orderof 0.1 eV in the density of states. Examination of the electronic structure reveals that thesystem is a semimetal with both electron and hole carriers. The existence of the pseudogapin the rope makes the conductivity and other transport properties of the metallic ropesignificantly different from those of isolated tubes, even without considering the effect oflocal disorder in low dimensions. Since the DOS increases rapidly away from the Fermilevel, the carrier density of the rope is sensitive to temperature and doping. The existenceof both electron and hole carriers leads to qualitatively different thermopower and Hall-effect behaviors from those expected for a normal metal. The optical properties of the ropeare also affected by the pseudogap. As illustrated by the calculated joint density of states(JDOS) in Fig. 20(b), there would be a finite onset in the infrared absorption spectrum for alarge perfectly ordered (10,10) rope, where one can assume k-conserving optical transitions.In the case of high disorder, an infrared experiment would more closely reflect the DOSrather than the JDOS. For most actual samples, the fraction of (10,10) carbon nanotubes(compared with other nanotubes of the same diameter) in the experimentally synthesizedropes appears to be small. However, the conclusion that broken symmetry induces a gapin the (n, n) tubes is a general result which is of relevance for tubes under any significant

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Figure E.20: (a) Calculated density of states for a rope of misaligned (10,10) carbon nan-otubes (broken line) and aligned tubes (solid line). The Fermi energy is at zero. (b)Calculated joint density of states for a rope of misaligned (broken line) and aligned (solidline) (10,10) tubes. Results are in units of states per meV per atom [83, 84]

Figure E.21: AFM image of a crossed SWNT device (A). Calculated structure of a crossed(5,5) SWNT junction with a force of 0 nN (B) and 15 nN (C) [86]

asymmetric perturbations, such as those due to structural deformations or external fields.

E.5.2 Crossed-tube junctions

The discussion of nanotube junctions in Sec. 3 is focused on the on-tube junctions, i.e.,forming a junction by joining two half tubes together. These systems are extremely inter-esting, but difficult to synthesize in a controlled manner at this time. Another way to formjunctions is to have two tubes crossing each other in contact [86]. (See Fig. 21.) This kindof crossed-tube junction is much easier to fabricate and control with present experimentaltechniques. When two nanotubes cross in free space, one expects that the tubes at theirclosest contact point will be at a van der Waals distance away from each other and thatthere will not be much intertube or junction conductance. However, as shown by Avourisand coworkers [87], for two crossed tubes lying on a substrate, there is a substantial forcepressing one tube against the other due to the substrate attraction. For a crossed-tubejunction composed of SWNTs with the experimental diameter of 1.4 nm, this contact forcehas been estimated to be about 5 nN [87]. This substrate force would then be sufficient todeform the crossed-tube junction and lead to better junction conductance.

In Fig. 21, panel A is an AFM image of a crossed-tube junction fabricated from two

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Figure E.22: Calculated conductance (expressed in units of e2/h) of a crossed (5,5) carbonnanotube junction with a contact force of 15 nN on a linear (top) and log (bottom) scale.The dashed (dotted-dashed) curve corresponds to the intra-tube (intertube) conductance[86]

single-walled carbon nanotubes of 1.4 nm in diameter with electrical contacts at each end[86]. Panels B and C show the calculated structure corresponding to a (5,5) carbon nanotubepressed against another one with zero and 15 nN force, respectively. Because of the smallerdiameter of the (5,5) tube, a larger contact force is required to produce a deformationsimilar to that of the experimental crossed-tube junction. The calculation was done usingthe ab initio pseudopotential density functional method with a localized basis [86]. As seenin panel C, there is considerable deformation, and the atoms on the different tubes are muchcloser to each other. At this distance, the closest atomic separation between the two tubesis 0.25 nm, significantly smaller than the van der Waals distance of 0.34 nm.

For the case of zero contact force (panel B in Fig. 21), the calculated intra-tube conduc-tance is virtually unchanged from that of an ideal, isolated metallic tube, and the intertubeconductance is negligibly small. However, when the tubes are under a force of 15 nN, thereis a sizable intertube or junction conductance. As shown in Fig. 22, the junction conduc-tance at the Fermi energy is about 5% of a quantum unit of conductance G0 = 2e2/h. Thejunction conductance is thus very sensitive to the force or distance between the tubes.

Experimentally, the conductance of various types of crossed carbon nanotube junc-tions has been measured, including metal-metal, semiconductor-semiconductor, and metal-semiconductor crossed-tube junctions. The experimental results are presented in Fig. 23.For the metal-metal crossed-tube junctions, a conductance of 2 to 6% of G0 is found, in goodagreement with the theoretical results. Of particular interest is the metal-semiconductorcase in which experiments demonstrated Schottky diode behavior with a Schottky barrierin the range of 200–300 meV, which is very close to the value of 250 meV expected fromtheory for nanotubes with diameters of 1.4 nm [86].

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Figure E.23: Current-voltage characteristics of several crossed SWNT junctions [86] (seetext)

E.5.3 Effects of Long-Range Disorder and External Perturbations

The effects of disorder on the conducting properties of metal and semiconducting carbonnanotubes are quite different. Experimentally, the mean free path is found to be muchlonger in metallic tubes than in doped semiconducting tubes [19, 20, 21, 24, 88]. This resultcan be understood theoretically if the disorder potential is long range. As discussed below,the internal structure of the wavefunction of the states connected to the sublattice structureof graphite lead to a suppression of scattering in metallic tubes, but not in semiconductingtubes. Figure 24 shows the measured conductance for a semiconducting nanotube deviceas a function of gate voltage at different temperatures. [88]. The diameter of the tube asmeasured by AFM is 1.5 nm, consistent with a single-walled tube. The complex structure inthe Coulomb blockade oscillations in Fig. 24 is consistent with transport through a number ofquantum dots in series. The temperature dependence and typical charging energy indicatesthat the tube is broken up into segments of length of about 100 nm. Similar measurementson intrinsic metal tubes, on the other hand, yield lengths that are typically a couple oforders of magnitude longer [19, 20, 21, 24, 88].

Theoretical calculations have been carried out to examine the effects of long-range ex-ternal perturbations [88]. In the calculation, to model the perturbation, a 3-dimensionalGaussian potential of a certain width is centered on one of the atoms on the carbon nan-otube wall. The conductance with the perturbation is computed for different Gaussianwidths, but keeping the integrated strength of the potential the same. Some typical tight-binding results are presented in Fig. 25. The solid lines show the results for the conductanceof a disorder-free tube, while the dashed and the dot-dashed lines are, respectively, for asingle long-range (σ=0.348 nm, ∆V = 0.5 eV) and a short-range (σ=0.116 nm, ∆V = 10 eV)scatterer. Here ∆V is the shift in the on-site energy at the potential center. The conduc-tion bands (i.e., bands crossing the Fermi level) of the metallic tube are unaffected by thelong-range scatterer, unlike the lower and upper subbands of both the metallic and semi-

conducting tubes, which are affected by boh long- and short-range scatterers. All subbandsare influenced by the short-range scatterer. The inset shows an expanded view of the onsetof conduction in the semiconducting tube at positive E, with each division corresponding

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Figure E.24: Conductance vs. gate voltage Vg for a semiconducting single-walled carbonnanotube at various temperatures. The upper insert schematically illustraates the samplegeometry and the lower insert shows dI/dV vs. V and Vg plotted as a gray scale [88]

to 1 meV. Also, the sharp step edges in the calculated conductance of the perfect tubes arerounded off by both types of perturbations.

Both the experimental and theoretical findings strongly suggest that long-range scatter-ing is suppressed in the metallic tubes. One can actually understand this qualitatively fromthe electronic structure of a graphene sheet [89, 90]. The graphene structure has two atomsper unit cell. The properties of electrons near the Fermi energy are given by those statesnear the corner of the Brillouin zone. (See Fig. 26.) If we look at the states near this pointand consider them in terms of a k-vector away from the corner K point, then they can bedescribed by a Dirac Hamiltonian. For these states, the wavefunctions can be written interms of a product of a plane wave component (with a vector k) and a pseudo-spin whichdescribes the bonding character between the two atoms in the unit cell. The interestingresult is that this pseudo-spin points along k. For example, if the state at k is bonding, thenthe state at −k is antibonding in character. Within this framework, one can work out thescattering between the allowed states in a carbon nanotube due to long-range disorder, i.e.,disorder with Fourier components V (q) such that q ¿ K. This, for example, will be thecase for scattering by charged trap states in the substrate (oxide traps). In this case, thedisorder does not couple to the pseudo-spin portion of the wavefunction, since the disorderpotential is approximately constant on the scale of the interatomic distance. The resultingmatrix element between states is then [89, 90]: |〈k′|V (r)|k〉|2 = |V (k− k′)|2 cos2[(1/2)θk,k′ ],where θk,k′ is the angle between the initial and final states. The first term in V (k − k′)is the Fourier component at the difference in k values of the initial and final envelopewavefunctions. The cosine term is the overlap of the initial and final spinor states.

For a metallic tube [Fig. 26(b)], backscattering in the conduction band corresponds toscattering between k and −k. Such scattering is forbidden, because the molecular orbitalsof these two states are orthogonal. In semiconducting tubes, however, the situation is quitedifferent [Fig. 26(c)]. The angle between the initial and final states is less than π, and scat-tering is thus only partially suppressed by the spinor overlap. As a result, semiconductingtubes should be sensitive to long-range disorder, while metallic tubes should not. However,short-range disorder which has Fourier components q ∼ K will couple the molecular orbitalstogether and lead to scattering in all of the subbands. These theoretical considerations agree

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Figure E.25: Tight-binding calculation of the conductance of a (a) metallic (10,10) tubeand (b) semiconducting (17,0) tube in the presence of a Gaussian scatterer. The energyscale on the abscissa is 0.2 eV per division in each graph [88]

Figure E.26: (a) Filled states (shaded) in the first Brillouin zone of a p-type graphene sheet.There are two carbon atoms per unit cell (lower right inset). The dispersions of the statesnear EF are cones whose vertices are located at the corner points of the Brillouin zone. TheFermi circle, defining the allowed k vectors, and the band dispersions are shown in (b) and(c) for a metallic and a semiconducting tube, respectively [88]

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well with experiment and with the detailed calculations discussed above. Long-range disor-der due to, e.g., localized charges near the tube, breaks the semiconducting tube into a seriesof quantum dots with large barriers, resulting in a dramatically reduced conductance anda short mean free path. On the other hand, metallic tubes are insensitive to this disorderand remain near-perfect 1D conductors.

E.6 Summary

This Chapter gives a short review of some of our theoretical understanding of the structuraland electronic properties of single-walled carbon nanotubes and of various structures formedfrom these nanotubes. Because of their nanometer dimensions, the nanotube structures canhave novel properties and yield unusual scientific phenomena. In addition to the multi-walled carbon nanotubes, single-walled nanotubes, nanotube ropes, nanotube junctions,and non-carbon nanotubes have been synthesized.

These quasi-one-dimensional objects have highly unusual electronic properties. For theperfect tubes, theoretical studies have shown that the electronic properties of the carbonnanotubes are intimately connected to their structure. They can be metallic or semicon-ducting, depending sensitively on tube diameter and chirality. Experimental studies usingtransport, scanning tunneling, and other techniques have basically confirmed the theoret-ical predictions. The dielectric responses of the carbon nanotubes are found to be highlyanisotropic in general. The heat capacity of single-wall nanotubes is predicted to have acharacteristic linear T dependence at low temperature.

On-tube metal-semiconductor, semiconductor-semiconductor, and metal-metal junctionsmay be formed by introducing topological structural defects, and these junctions have beenshown to behave like nanoscale device elements. For example, different half-tubes maybe joined with 5-member ring/7-member ring pair defects to form a metal-semiconductorSchottky barrier. The calculated electronic structure of these junctions is very similar tothat of standard metal-semiconductor interfaces, and in this sense, they are molecular leveldevices composed of the single element, carbon. Recent experimental measurements haveconfirmed the existence of such Schottky barrier behavior in nanotube ropes and acrosskinked nanotube junctions. Similarly, 5-7 defect pairs in different carbon and non-carbonnanotubes can produce semiconductor-semiconductor and metal-metal junctions. The ex-istence of metal-metal nanotube junctions in which the conductance is suppressed for sym-metry reasons has also been predicted. Thus, the carbon nanotube junctions may be usedas nanoscale electronic elements.

The influence of impurities and local structural defects on the conductance of carbonnanotubes has also been examined. It is found that local defects in general form well de-fined quasi-bound states even in metallic nanotubes. These defect states give rise to peaksin the LDOS and reduce the conductance at the energy of the defect levels by a quan-tum unit of conductance via resonant backscattering. The theoretical studies show that,owing to the unique electronic structure of the graphene sheet, the transport propertiesof (n, n) metallic tubes appear to be very robust against defects and long-range pertur-bations near EF . Doped semiconducting tubes are much more susceptible to long-rangedisorder. These results explain the experimental findings of the long coherence length inmetallic tubes and the large difference in mean free path between the metallic and dopedsemiconducting tubes. For nanotube ropes, intertube interactions are shown to alter the

251

electronic structure of (n, n) metallic tubes because of broken symmetry effects, leading toa pseudogap in the density of states and to semimetallic behavior. Crossed-tube junctionshave also been fabricated experimentally and studied theoretically. These systems showsignificant intertube conductance for metal-metal junctions and exhibit Schottky behaviorfor metal-semiconductor junctions when the tubes are subjected to contact force from thesubstrate.

The carbon nanotubes are hence a fascinating new class of materials with many uniqueand desirable properties. The rich interplay between the geometric and electronic structureof the nanotubes has given rise to many interesting, new physical phenomena. At thepractical level, these systems have the potential for many possible applications.

252

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257

Appendix F

Low dimensional systems as

promising thermoelectric

materials: a study of

one-dimensional Bismuth

Nanowires

References:

• T. M. Tritt, Recent trends in Thermoelectric Materials III, Vol. 71 of Series of Semi-conductors and Semimetals, Academic Press, Chapter 1.

• Z. Zhang, X. Sun, M. S. Dresselhaus, J. Y. Ying, and J. Heremans, Appl. Phys.

Lett., 73, 1589 (1998)

• Y. -M. Lin and X. Sun and M. S. Dresselhaus, Phys. Rev. B, 62, 4610 (2000)

• Z. Zhang, X. Sun, M. S. Dresselhaus, J. Y. Ying and J. Heremans, Phys. Rev. B, 61,4850 (2000).

F.0.1 Introduction to thermoelectricity

Evaluation of new materials including low dimensional materials for thermoelectric applica-tions is usually made in terms of the dimensionless thermoelectric figure of merit ZT whereT is the temperature (in degrees Kelvin) and Z is given by

Z =S2σ

κ, (F.1)

where S is the thermoelectric power or Seebeck coefficient, σ is the electrical conductivityand

κ = κe + κph, (F.2)

258

0 1 2 3 4 5dW (nm)

0

20

40

60

80

100

S2 n

(1023

µV2 cm

−3 K

−2 )

Theory

Figure F.1: The comparison between experimental data for S2n vs quantum well widthdW and the theoretical curve at optimal doping level to maximize Z2DT for the optimumthermoelectric figure of merit for a strain relaxed Si/Si0.7Ge0.3 quantum well superlatticeof 15 periods at room temperature.

is the thermal conductivity, which includes contributions from carriers (κe) and from thelattice (κph). Equation (F.1) emphasizes the importance of a large S for high thermoelectricperformance (or high ZT ), where S denotes the voltage generated by a thermal gradient.Large values of ZT require high S, high σ, and low κ. Since an increase in S normallyimplies a decrease in σ because of carrier density considerations, and since an increase in σimplies an increase in the electronic contribution to κ as given by the Wiedemann–Franz law,it is very difficult to increase Z in typical thermoelectric materials. The best commercial3D thermoelectric material is Bi0.5Sb1.5Te3 in the Bi2(1−x)Sb2xTe3(1−y)Se3y family with aroom temperature ZT ≈ 1. It is believed that if materials with ZT ≈ 3 could be developed,many more practical applications for thermoelectric devices would follow.

F.1 Proof-of-Principle Studies

Early studies of low dimensional thermoelectricity focused on the demonstration of proof-of-principle of enhanced ZT within a quantum well structure using simple theoretical models,comparisons between theory and experiment, and comparisons between the low dimensional(2D) and 3D Seebeck coefficient, all comparisons being carried out under optimum dopingconditions.

The demonstration of proof-of-principle in the strain relaxed Si/Si0.7Ge0.3 superlatticesystem is shown in Fig. F.1. For the proof-of-principle studies, superlattices were grownwith 15 superlattice periods, having quantum well widths between 10–50 A alternating with300 A of Si0.7Ge0.3 of barrier layer, and the measured S2n at 300 K were compared to modelcalculations based on the well-established band structures of Si and Si1−xGex alloys, usingonly literature values and no adjustable parameters. This comparison between the modelcalculation and measurements for S2n provides clear confirmation that the reduction of thesize of the quantum well results in an increase in S2n and that the model calculation has asimilar dependence on quantum well width as the experimental points, when taking account

259

Table F.1: Bismuth parameters

Property Bulk Trigonal Binary Bisectrix

Mass Density (g/cm3) (300 K) 9.8Melting Point (K) (1 atm) 544.4Velocity of sound (105 cm/s) (4.2 K) 2.02 2.62 2.7Velocity of sound (105 cm/s)(300 K) 1.972 2.540 2.571Phonon mean free patha (nm) (77 K) 14.7 15.2 14.8Thermal conductivity (300 K) (W/mK) 6.0 9.8 9.8Lattice constantb (A) c = 11.862 a = 4.5460 a = 4.5460Compressibility (Mbar−1) 1.82 0.62 0.62Bulk modulus (Mbar−1) 0.326 × 10−3

Young’s modulus (dyn/cm) 2.12 × 1011 3.10 × 1011 3.10 × 1011

Volume coeff of thermal expansion (K−1) 3.965 × 10−5

L-point band gap (meV) (0 K) 13.6 13.6 13.6Plasma frequency (2 K) (cm−1) 158 ± 3Work function (eV) 4.22Debye temperature (K) 112Static dielectric constant 84 105 105Carrier density (77 K)c (1017 cm−3) 4.4Carrier density (300 K) (1017 cm−3) 2.4

aFrom the relation κ = Cvvs`/3. Along [0112] and [1011], ` = 15.3 nm and 15.7 nm, respec-tively.bThe rhombohedral angle for Bi is α = 57◦14.2′ and sublattice parameter u = 0.237 ascompared to α = 60◦ and u = 0.25 for a cubic system. For Sb the lattice constants area = 4.308 A, c = 11.274 A, α = 57◦6.5′ and u = 0.233.cThe carrier density at 4 K for Bi is 2.7× 1017/cm3, for Sb is 3.7× 1019/cm3, and for As is2 × 1020/cm3.

of experimental uncertainties in the data.

F.2 Basic properties of Bi

Bismuth is a very attractive material for low-dimensional thermoelectricity because of thevery large anisotropy of the three ellipsoidal constant energy surfaces for electrons at theL-point in the rhombohedral Brillouin zone (see Fig. F.2), and the high mobility of thecarriers for the light mass electrons. In addition, bulk bismuth has carriers with very longmean free paths for electronic transport and heavy mass ions which are highly effectivefor scattering phonons (see Table F.1). Bismuth can also be alloyed isoelectronically withantimony to yield a high mobility alloy with highly desirable thermoelectric properties.

As a bulk material, semimetallic Bi has a low Seebeck coefficient S because of theapproximate cancellation between the contributions to S from the electron and hole carriers,

260

Table F.2: The band structure parameters of bulk Bi at T ≤ 77 K

Parameters Notation Value

Band Overlapa ∆0 –38 meV

Band Gap at L-point EgL 13.8 meV

Electron Effective me1 0.00119 m0

Mass Tensor Elements me2 0.263 m0

at the Band Edge for me3 0.00516 m0

L(A) pocketb me4 0.0274 m0

T -point Hole Effective mh1 0.059 m0

Mass Tensor Elements mh2 0.059 m0

at the Band Edge mh3 0.634 m0

Electron Mobilityc µ1 68.0at 77 K µ2 1.6in units µ3 38.0×104 cm2/Vs µ4 –4.3

Hole Mobilityc at 77 K µh1 = µh2 12.0in units ×104 cm2/Vs µh3 2.1

aThe band overlap for Sb is 17.75 meV and for As is 356 meV.bThe tilt angle of the L-point electron ellipsoids are 6.0◦, −4◦, −4◦ for Bi, Sb, and As,respectively. The tilt angle of the H-point hole “ellipsoids” are 53◦ for Sb and 37.5◦ for themajor As hole ellipsoid.cThe form of the effective mass tensor and the mobility tensor are assumed to be the samefor a given carrier pocket.

261

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� � � "�#�$�%�&'� (��

��� $�%�� ) � *+�

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Figure F.2: The Fermi surfaces of Bi, showing the Brillouin zone with the fifth-band holepocket about the T -point and the three sixth-band L-point electron pockets labeled A, B,and C. For quantum wells with their confinement direction, or for nanowires having theirwire axes in the bisectrix-trigonal plane, the mirror plane symmetry of the bulk bismuthstructure results in the crystallographic equivalence of the L-point carrier pockets B andC. However, the L-point carrier pocket A is not equivalent crystallographically to carrierpockets B or C.

since for a two carrier system, the Seebeck coefficient can be written as

S =σeSe + σhSh

σe + σh(F.3)

where σe, σh, Se and Sh respectively, denote the electrical conductivity and Seebeck coef-ficient for the electrons and holes. For the case of bismuth, the mobility of the electrons ismuch larger that of the holes, so that S tends to be weakly negative. It was early recognizedthat Bi could be a good thermoelectric material if the hole carriers could be removed; how-ever, no reliable mechanism was proposed to remove the hole carriers in pure bismuth. Onthe other hand, it was recognized long ago that Bi1−xSbx alloys when properly doped andoriented could be among the best presently available thermoelectric materials, especially inthe liquid nitrogen temperature range (near 77 K).

Low dimensionality, however, offers an opportunity to overcome the problem of the lowSeebeck coefficient in pure bismuth. As the quantum well (or wire) width decreases, theband edge for the lowest subband in the conduction band rises above that for the highestsubband in the valence band, thereby inducing a semimetal-semiconductor transition. If the2D (or 1D) bismuth system is then doped to the optimum doping level, a large enhancementin Z2DT (and even more enhancement in Z1DT ) should be possible as the quantum well (orwire) width is decreased.

Table F.2 summarizes the band structure parameters of bulk Bi, and their temperaturedependence is given in Table F.3. The modeling of Bi nanowires and the prediction of theirthermoelectric properties will be based on these band structure parameters.

262

Table F.3: Temperature dependence of selected bulk Bi band structure parameters.

Parameters Temperature Dependence

Band Overlap(meV)

∆0 =

−38 (meV) (T < 80K)

−38 − 0.044(T − 80)+4.58 × 10−4(T − 80)2 (T > 80K)−7.39 × 10−6(T − 80)3

(F.4)

Direct Band Gap(meV)

EgL = 13.6 + 2.1 × 10−3T + 2.5 × 10−4T 2 (F.5)

L-pointElectron EffectiveMass Components

[me(T )]ij =[me(0)]ij

1 − 2.94 × 10−3T + 5.56 × 10−7T 2(F.6)

F.3 Nanowires

F.3.1 Introduction to Nanowires

Thus far, bismuth is the dominant thermoelectric quantum wire material that has been fab-ricated and studied for thermoelectric applications, though some success has been demon-strated with the fabrication of quantum wires from antimony, Bi2Te3, and Si. However,except for the case of Bi and Sb, little attention has been given to the thermoelectric proper-ties of these wires. Ballistic transport in thin metallic wires has been studied more generallyfor many years. In this section, the structure, characterization and thermoelectric-relatedproperties of bismuth nanowires is reviewed.

F.3.2 Structure and Synthesis of Bismuth Nanowires

Arrays of hexagonally-packed parallel bismuth nanowires, 7–110 nm in diameter and 25–65 µm in length, have been prepared. These nanowires are embedded in a dielectric matrixof anodic alumina, which, because of its array of parallel nano-channels, is used as a templatefor preparing the Bi nanowires (see Fig. F.3). The bismuth is confined to these nanochannelsand the bismuth does not diffuse into the anodic alumina matrix. The Bi nanowires arehighly oriented with a common crystallographic direction along the wire axis. Since the bandstructure of Bi is highly anisotropic, the transport properties of Bi nanowires are expected tobe dependent on the crystallographic orientation along the wire axes. In addition, structuralanalysis shows that the crystal structure of bulk Bi is maintained in the nanowires, indicatingthat many properties of bulk Bi may be utilized in modeling the behavior of Bi nanowires.

263

Figure F.3: Cross-sectional view of the cylindrical channels of 65 nm average diameter of ananodic alumina template, shown as a transmission electron microscope (TEM) image. Thetemplate has been mostly filled with bismuth, and the TEM image was taken after the topand bottom sides of the sample had been ion milled with 6 KV Ar ions.

F.3.3 Electronic Structure of Nanowires

To model the electronic structure, we assume, as a first approximation, the simplest possiblemodel for an ideal 1D quantum wire, where the carriers are confined inside a cylindricalpotential well bounded by a barrier of infinite potential height. An extension of this simpleapproach provides a reasonable approximation for a Bi nanowire embedded in an aluminatemplate, in view of the large band gap of the anodic alumina template (3.2 eV), whichprovides excellent carrier confinement for the embedded quantum wires. Due to the smallelectron effective mass components of Bi, the quantum confinement effects in Bi nanowiresare more prominent than for other wires with the same diameter.

Since the electron motion in the quantum wires is restricted in directions normal to thewire axis, the confinement causes the energies associated with the in-plane motion to bequantized, and the energy of the lowest energy subband will be raised by approximately

∆E ∼ h2

m∗pd

2W

(F.7)

where m∗p is the in-plane effective mass of the electrons and dW is the wire diameter. Since

electron motion is only allowed along the wire axis, the electrons are expected to behavelike a 1D electron system, with a dispersion relation that has the form

Enm(kl) = εnm +h2kl

2

2m∗l

(F.8)

where εnm represents a quantized energy level labeled by two quantum numbers (n, m), kl

is the wavenumber of the electron wavefunctions traveling along the wire axis, and m∗l is the

264

dynamical effective mass for electrons moving along the wire. For materials with a highlyanisotropic electronic energy band structure such as Bi, the mass m∗

p which determinesthe subband energies εnm can be very different from the mass m∗

l which characterizes themotion along the wire.

In the nanowire system, the quantized subband energy εnm and the transport effectivemass m∗

l along the wire axis are the two most important band parameters that determinealmost every electronic property of this unique 1D system. However, due to the highlyanisotropic electron and hole pockets, the calculation of the band structure in Bi nanowireshas been very challenging.

Recently, calculations of the band structure of 1D Bi quantum wires have been carriedout, which explicitly take into account the cylindrical wire boundary conditions and theanisotropic carrier effective masses in Bi. In addition, the non-parabolic features of theL-point conduction band and the temperature dependence of the various band parametersare also included to provide a more accurate model for the electronic structure of the Binanowires.

For an infinitely long circular wire with a diameter dW , the z′ axis is taken to beparallel to the wire axis with the x′ and y′ axes lying on the cross-sectional plane of thewire. Since the wires are allowed to be oriented along an arbitrary direction with respect tothe crystallographic directions, the inverse effective mass tensor of one of the carrier pocketsin the wire coordinates (x′, y′, z′) has the general form

α ≡ M−1 =

α11 α12 α13

α21 α22 α23

α31 α32 α33

(F.9)

where αij = αji. Explicit values for the components of the effective mass tensor M forthe T -point holes and L-point electrons are given in Table F.2. Since there is only onecarrier pocket for holes at the T point, Eq. (F.9) is sufficient for describing the T -pointholes. However, for the L-point electrons, Eq. (F.9) describes only carrier pocket A (seeFig. F.2), and rotations of Eq. (F.9) by ±2π/3 around the trigonal axis are needed to obtainthe effective mass tensors for the B and C electron carrier pockets.

For the T -point holes, parabolic energy bands are assumed, and the Schodinger equationis simplified and given by

− h2

2

(

α11∂2

∂x′2 + α22∂2

∂y′2

)

u =

(

E − h2k2z′

2m33

)

u. (F.10)

Equation (F.10) has solutions u(x′, y′) that satisfy the boundary condition u(r′ = dW /2) =0, yielding the eigenvalues of u(x′, y′) in Eq. (F.10) that are quantized

Enm(kz′) = εnm +h2k2

z′

2m33, (F.11)

where εnm is the eigenvalue of Eq. (F.10) corresponding to the band edge eigenstate atkz′ = 0 labeled by the quantum numbers (n, m). Here

m33 = z′ · M · z′ (F.12)

265

Table F.4: Calculated effective mass components of each carrier pocket for determining theband structure of Bi nanowires at 77 K along the indicated crystallographic directions, basedon the effective mass parameters of bulk bismuth given in Table F.2. The z ′ direction ischosen along the wire axis. Calculations are done for Bi nanowires oriented along the threeprincipal axes (trigonal, binary, and bisectrix) and the [0112] and [1011] directions (whichare preferential growth directions for Bi nanowires) and correspond to (0,0.8339,0.5519) and(0,0.9503,0.3112) in Cartesian coordinates, respectively. All mass values in this table are inunits of the free electron mass, m0.

Mass Component Trigonal Binary Bisectrix [0112] [1011]

mx′ 0.1175 0.0023 0.0023 0.0029 0.0024e− pocket A my′ 0.0012 0.2659 0.0012 0.0012 0.0012

mz′ 0.0052 0.0012 0.2630 0.2094 0.2542

mx′ 0.1175 0.0023 0.0023 0.0016 0.0019e− pocket B my′ 0.0012 0.0016 0.0048 0.0125 0.0071

mz′ 0.0052 0.1975 0.0666 0.0352 0.0526

mx′ 0.1175 0.0023 0.0023 0.0016 0.0019e− pocket C my′ 0.0012 0.0016 0.0048 0.0125 0.0071

mz′ 0.0052 0.1975 0.0666 0.0352 0.0526

mx′ 0.0590 0.6340 0.6340 0.1593 0.3261hole pocket my′ 0.0590 0.0590 0.0590 0.0590 0.0590

mz′ 0.6340 0.0590 0.0590 0.2349 0.1147

266

is the effective mass component along the wire axis, and the in-plane effective masses are

mx′ ≡ α−111 = (x′ · M−1 · x′)−1

my′ ≡ α−122 = (y′ · M−1 · y′)−1 (F.13)

in the x′ and y′ directions, respectively. The eigenvalue εnm has an analytic expression onlywhen α11 = α22, but in general, the values of εnm must be solved numerically. This is truefor both the T -point holes and for the L-point electrons.

From the band structure parameters for bulk Bi (Table F.2), values for mx′ , my′ , andmz′ (or mz) at 77 K for Bi nanowires for the three principal crystallographic axes (trigonal,binary, and bisectrix directions), and for the preferential [0112] and [1011] growth directionsare given in Table F.4. It should be noted that mx′ and my′ can be interchanged withoutaffecting any physical results.

The calculated subband structure of Bi quantum wires oriented along the [0112] direc-tions are shown in Fig. F.4 at 77 K. For the [0112] wires, the degeneracy at the L point islifted, resulting in two inequivalent groups of carrier pockets: a single electron pocket A andtwo electron pockets B, C with the same symmetry and band parameters as each other butdifferent from pocket A. The L-point electron pocket A has smaller mass components (mx′ ,my′) in the quantum confined direction than the electron pockets B and C (see Table F.4).Therefore, the electron pocket A forms a higher energy conduction subband, while the elec-tron pockets B and C form a two-fold degenerate subband at a lower energy (see Fig. F.4).It should be pointed out that since electron pocket A has a larger mass component (mz′)along the wire axis than pockets B and C, the dispersion relation of the L(A) subband hasa smaller curvature (see Fig. F.4). The band edge of the lowest subband of the L(B, C)electrons increases with decreasing wire diameter dW , while the highest subband edges ofthe T -point and L-point holes move downwards in energy. At dW < 49.0 nm, the energyof the lowest L-point conduction subband edge exceeds that of the highest T -point valencesubband edge, indicating that these nanowires become semiconducting.

F.3.4 Doping of Bi nanowires

For intrinsic Bi nanowires, equal numbers of electrons and holes are expected, whetherin the semimetallic or semiconducting state. However, in thermoelectric applications ofbismuth, it is necessary to control the Fermi level so that: (1) the transport phenomena aredominated by a single type of carrier only, i.e., electrons or holes, (2) the electrochemicalpotential is placed to achieve the optimum ZT . In addition, to optimize the efficiencyof thermoelectric devices, it is essential to obtain a high Seebeck coefficient S. However,since the contributions from holes and from electrons have opposite signs with regard tothe Seebeck coefficient, the magnitude of S in pure Bi is usually very small, althoughindividual contributions from electrons or holes can be quite significant. Therefore, it isexpected that Bi nanowires can be a very promising thermoelectric material if the Fermilevel can be adjusted properly so that only electrons or only holes contribute to S and theelectrochemical potential is set to maximize ZT (see §F.2).

Since Bi is a group V element, the Fermi level can be increased by introducing a smallamount of group VI element, such as Te, which acts as an electron donor in Bi. Group IVelements such as Sn or Pb, on the other hand, act as electron acceptors in Bi, and can beused to synthesize p-type Bi.

267

0 50 100 150 200Wire diameter (nm)

−40

−20

0

20

40

60

80

100E

nerg

y (m

eV)

49.0 nm

T holesL e

−(A)

L e−(B,C)

L holes

−∆0 = 38

EgL = 13.6

Figure F.4: The subband structure at 77 K of Bi quantum wires oriented along the [0112]growth direction, showing the energies of the highest subbands for the T -point hole carrierpocket, the L-point electron pockets (A, B and C) as well as the L-point holes. The zeroenergy refers to the conduction band edge in bulk Bi. As the wire diameter dW decreases,the conduction subbands move up in energy, while the valence subbands move down. Atdc = 49.0 nm, the lowest conduction subband edge formed by the L(B, C) electrons crossesthe highest T -point valence subband edge, and a semimetal-semiconductor transition occurs.

268

We assume that the electronic band structure of Te-doped Bi is the same as pure Bi inthe spirit of a rigid band approximation, except for the presence of a Te donor energy levellocated below the conduction band edge by Ed.

The ionization energy Ed required for releasing donor electrons from Te atoms can beestimated using the Bohr hydrogen-like model,

Ed ' 13.6 × m∗/m0

ε2(eV) (F.14)

where m∗ is the effective mass at the conduction band edge, m0 is the free electron mass,and ε is the dielectric constant (ε ' 100) of Bi in the low frequency limit (see Table F.1).The value for the effective mass m∗ is taken as the average value of the electron effectivemass tensor, and at low temperatures, m∗/m0 ' 0.002. With this large value for ε andsmall value for m∗, we obtain a very small value for Ed

Ed ' 3.20 × 10−3meV (F.15)

with a very large effective Bohr radius

a∗0 = 0.5 (A) × ε

m∗ ' 2.5µm, (F.16)

so that the energy required to ionize the Te atom in Bi is very small. However, it should benoted that the results obtained in Eqs. (F.15) and (F.16) are only valid for bulk materials.In Te-doped Bi nanowires, where the wire diameter dW is smaller than the effective Bohrradius a∗0, the ionization energy will be increased due to the confinement of donor electronsto be close to the impurity atom. Since the Coulomb potential energy is inversely propor-tional to the distance between two charged particles, a rough estimation of the ionizationenergy Ed(1D) in nanowires can be made by multiplying the ionization energy Ed(3D) in bulkmaterials [Eq. (F.15)] by the ratio between the effective Bohr radius and the wire diameter.Thus, for 40 nm Te-doped Bi nanowires, an estimate for the ionization energy would be

Ed(1D) ' Ed(3D) ×a∗0dW

' 0.2 meV. (F.17)

At very low temperatures where the thermal energy kBT ≤ Ed(1D), the donor electrons willfreeze out, and the freeze-out temperature is Tfo ' 2 K for 40 nm Te-doped Bi nanowires.For the temperatures of interest (ranging from 4 K to 300 K), we can assume that all thedonor atoms are ionized in 40 nm Te-doped Bi nanowires and that each Te atom donatesone electron to the conduction band.

F.3.5 Semi-Classical Transport Model for Bi Nanowires

The thermoelectric-related transport coefficients of Te-doped Bi nanowires can be derivedfrom the simple semi-classical model, which is based on the Boltzmann transport equation.

We define the transport-related quantities L(α)1D as

L(α)1D = e2

∫8dk

π2d2W

(

− df

dE

)

τ(k)v(k)v(k)[E(k) − Ef ]α, (F.18)

269

in which E(k) is the electronic dispersion relation, τ(k) is the relaxation time, Ef is theFermi energy, and f(E) is the Fermi–Dirac distribution function

f(E) =1

1 + e(E−Ef )/kBT. (F.19)

Since the numerical calculation of Eq. (F.18) requires knowledge of the k dependence ofthe relaxation time τ(k), and since the calculation of τ(k) from fundamental principles ofscattering mechanisms is usually very complicated, we use a simple first approximation,known as the constant relaxation time approximation, to simplify the calculations of thethermoelectric properties of materials. In this formalism, τ(k) = τ is taken to be constantin k, and in energy, and τ can be related to the carrier mobility µ along the wire by

µ =eτ

m∗ (F.20)

where m∗ is the transport effective mass along the wire, and µ can, in principle, be obtainedfrom experimental measurements. Thus, the integration of Eq. (F.18) can be carried outreadily as long as the dispersion relation E(k) is known. For a one-band system describedby a parabolic dispersion relation, Eq. (F.18) becomes

L(0)1D = D

[1

2F− 1

2

]

(F.21)

L(1)1D =

(kBT )D[

32F 1

2

− 12ζ∗F− 1

2

]

(for electrons)

−(kBT )D[

32F 1

2

− 12ζ∗F− 1

2

]

(for holes)(F.22)

L(2)1D = (kBT )2D

[5

2F 3

2

− 3ζ∗F 1

2

+1

2ζ∗2F− 1

2

]

(F.23)

where D is given by

D =16e

πd2W

(2m∗kBT

h2

) 1

2

µ, (F.24)

and Fi denotes the Fermi–Dirac related functions which is given by

Fi =

∫ ∞

0

xidx

exp(x − ζ∗) + 1, (F.25)

with fractional indices i = − 12 , 1

2 , 32 , . . .. The reduced chemical potential ζ∗ is defined as:

ζ∗ =

{

(Ef − ε(0)e )/kBT (for electrons)

(ε(0)h − Ef )/kBT (for holes).

(F.26)

where ε(0)e and ε

(0)h are the band edges for electrons and holes, respectively.

For Bi quantum wire systems, there are many 1D subbands due to the multiple carrierpockets at the L points and the T point, and the quantum confinement-induced bandsplitting also forms a set of 1D subbands from each single band in bulk materials. Therefore,when considering the transport properties of real 1D nanowire systems, contributions fromall of the subbands near the Fermi energy should be included. In a multi-band system, the

270

L(α)’s, should be replaced by the sum L(α)total =

∑

i L(α)i of contributions from each subband

(label by i), and the transport coefficients σ, S, and κe then become

σtotal =∑

i

σi (F.27)

Stotal =

∑

i σiSi∑

i σi(F.28)

κe,total =1

e2T

[

L(2)total −

(L(1)total)

2

L(0)total

]

(F.29)

where σi and Si are the electrical conductivity and the thermopower corresponding to eachsubband, respectively. In Bi nanowires, the sums in Eqs. (F.27)–(F.29) include subbandsassociated with electron pockets A and (B,C) as well as contributions from T -point holesand L-point holes (see §F.3.3).

Another physical quantity of interest in thermoelectric applications is the lattice thermalconductivity κL which, together with electronic thermal conductivity κe, determines thetotal thermal conductivity of the system. From kinetic theory, the thermal conductivity ofphonons is given by

κL =1

3Cvv` (F.30)

where Cv is the heat capacity per unit volume, v is the sound velocity, and ` is the meanfree path for phonons. We note that, for an ideal quantum wire system embedded in ahost material with a large band gap, the electron wavefunctions are well confined withinthe quantum wire, and they can only travel along the wire axis. However, the host materialthat confines electrons cannot confine the phonon paths, and thus, because of acousticmismatch, phonons will be scattered when they move across the wire boundary. Thisincreased boundary scattering of phonons in the quantum wire system will decrease thephonon mean free path ` as well as the lattice thermal conductivity along the wire. Thesimplest approximation to model the lattice thermal conductivity in the quantum wiresystem is to replace the phonon mean free path ` in Eq. (F.30) by the wire diameter dW

if dW < ` in the bulk material. It should be noted that for dW ¿ `, the lattice thermalconductivity is expected to decrease dramatically, more so than the decrease in the electricalconductivity. This reduction in the lattice thermal conductivity is one of the reasons forthe expected enhanced thermoelectric performance in low-dimensional systems.

F.3.6 Optimization of Z1DT through doping at 77 K

In applying the general results of §F.3.5 to calculate Z1DT for Bi nanowires, the anisotropyof the electronic structure results in anisotropic effective mass tensors, and other relatedquantities have to be considered. For example, the mobility tensor for each carrier pocket forBi is also highly anisotropic. For the L-point electron pocket A (see Fig. F.2), the mobilitytensor has the form

µe(A) =

µe1 0 00 µe2 µe4

0 µe4 µe3

, (F.31)

271

Table F.5: Values of the mobility tensor elements for electron and hole pockets of Bi at77 K. The mobility values are given in units of cm2V−1s−1.

µe1 µe2 µe3 µe4 µh1 µh3

6.8×103 1.6×104 3.8×103 -4.3×104 1.20×105 2.1×104

and the mobility tensors for electron pockets B and C can be derived by a rotation of µe(A)

by ±120◦ about the trigonal axis. For the T -point holes, the mobility tensor has the form

µh =

µh1 0 00 µh1 00 0 µh3

. (F.32)

The values for these mobility tensor elements at 77 K are listed in Table F.5. Since themobility tensors are anisotropic, the mobility µl for carriers traveling along the wire willdepend on the wire orientation, and µl is given by

µl = (l · µ−1 · l)−1 (F.33)

where l is the unit vector along the wire axis, which follows from the general definition ofthe carrier mobility in terms of µ = eτ/m∗ and from Matthiessen’s rule summing 1/τi foreach scattering process i.

The lattice thermal conductivity in bulk Bi is also anisotropic, and has the form

κL =

κL,⊥ 0 00 κL,⊥ 00 0 κL,‖

(F.34)

where κL,‖ and κL,⊥ are the thermal conductivities parallel and perpendicular to the trigonalaxis, respectively. By extrapolating the experimental data for κL measured between 100 Kand 300 K, the lattice thermal conductivity tensor elements at 77 K are estimated as κL,⊥ =13.2 (W/mK) and κL,‖ = 9.9 (W/mK), respectively. For Bi nanowires oriented in directionsother than the three principal axes, the lattice thermal conductivity along the wire is thengiven by

κL,l = l · κL · l= cos2 θκL,⊥ + sin2 θκL,‖ (F.35)

where θ is the angle between the wire axis and the trigonal axis.The phonon mean free paths ` in Bi nanowires are estimated by the heat capacity Cv,

sound velocities v and the thermal conductivity κL,l via Eq. (F.30). At 77 K, the value

for the heat capacity of Bi is measured as Cv ' 1.003 (JK−1cm−3). The measured soundvelocities v of Bi at 1.6 K and 300 K along selected directions are listed in Table F.3.6, inwhich the interpolated values for v at 77 K are also given. The calculated phonon meanfree path ` of bulk Bi at 77 K is listed in Table F.1 for Bi crystals oriented along the three

272

Table F.6: The sound velocities v of Bi along the three principal axes. The values of 77 Kare interpolated from the experimentally measured results at 1.6 K and 300 K.

v (105 cm/s)T (K) Trigonal Binary Bisectrix (0, 1√

2, 1√

2) [0112] [1011]

1.6 2.02 2.62 2.70 2.15 2.26 2.58

77 2.01 2.60 2.67 2.13 2.24 2.45

300 1.972 2.540 2.571 2.082 2.18 2.375

principal axes, and also along the [1011], and the [0112] directions. As the wire diametersbecome smaller than the phonon mean free path calculated in bulk Bi, the scattering at thewire boundary becomes the dominant scattering process for phonons, and the phonon meanfree path in these nanowires is approximately limited by the wire diameter, or l ' dW . Thus,the lattice thermal conductivity κL in Te-doped Bi nanowires will decrease significantly asthe wire diameter decreases below dW ≤ 15 nm (see Table F.1).

Using the general formalism presented in §F.3.5 for S, σ, κe, and the above discussionon κL and procedures to account for the multiple carrier pockets and their anisotropy, thethermoelectric figure of merit Z1DT has been calculated. Figure F.5 shows the calculatedZ1DT for n-type Bi nanowires oriented along the trigonal axis at 77 K as a function of thedopant concentrations for three different wire diameters. We note that the value of Z1DTfor a given donor concentration increases drastically with decreasing wire diameter dW , andthe maximum Z1DT for each wire diameter occurs at an optimized donor concentrationNd(opt) which increases somewhat as the wire diameter decreases. For 5 nm Bi nanowiresoriented along the trigonal axis at 77 K, the maximum Z1DT at 77 K is about 6, with anoptimized electron concentration Nd(opt) ' 1018 cm−3. The value of Z1DT also stronglydepends on the wire orientation due to the anisotropic nature of the Bi band structure andof the thermal properties of Bi. Figure F.6 shows the calculated figure of merit Z1DT at77 K as a function of donor concentration Nd for 10 nm Bi nanowires oriented in differentdirections. For 10 nm Bi nanowires at 77 K, the trigonal nanowires have the highest optimalZ1DT which is about 2.0, while bisectrix wires have the lowest optimal Z1DT ' 0.4. Theoptimum carrier concentrations Nd(opt) and the corresponding Z1DT of n-type Bi nanowiresat 77 K are listed in Table F.7 for various wire diameters and orientations. Figure F.7 showsthe calculated optimal Z1DT at 77 K as a function of wire diameter for n-type Bi nanowiresoriented along the three principal axes, and the [1011], and the [0112] growth directions.

As a comparison, the optimum acceptor concentration Na(opt) and the correspondingZ1DT for p-type Bi nanowires are calculated and listed in Table F.7 for various wire diam-eters and orientations. Compared with the results in Table F.7 for n-type Bi nanowires, wenote that p-type Bi nanowires in general have a much lower Z1DT .

F.3.7 Temperature-Dependent Resistivity of Bi Nanowires

Measurements of the temperature dependence of the resistance R(T ) of Bi nanowire ar-rays have been carried out on samples prepared both from the liquid phase by pressureinjection and by vapor phase deposition, yielding results consistent with each other. Due

273

1016

1017

1018

1019

1020

Dopant Concentration (cm−3

)

0

1

2

3

4

5

6

7

Z1D

T

5 nm

10 nm

40 nm

Figure F.5: Calculated Z1DT for Te-doped Bi nanowires oriented along the trigonal axis at77 K as a function of Te dopant concentration for three different wire diameters.

Table F.7: The optimum dopant concentrations Nd(opt) (in 1018 cm−3) and the correspond-ing Z1DT of n-type and p-type Bi nanowires at 77 K for various wire diameters and orien-tations.

Wire 5 nm 10 nm 40 nmOrientation Nd(opt) Z1DT Nd(opt) Z1DT Nd(opt) Z1DT

Trigonal (n-type) 0.96 6.36 0.81 2.0 0.38 0.31(p-type) 0.96 6.36 12.9 0.72 6.2 0.17

Binary (n-type) 0.35 3.68 0.28 1.14 0.56 0.13(p-type) 0.79 1.78 10.3 0.16 7.9 0.05

Bisectrix (n-type) 4.1 2.21 1.78 0.40 4.97 0.03(p-type) 0.74 0.32 0.19 0.40 0.50 0.07

[1011] (n-type) 3.21 2.69 1.57 0.51 2.57 0.04(p-type) 1.04 1.16 0.43 0.19 0.63 0.05

[0112] (n-type) 2.07 3.41 1.33 0.70 2.73 0.06(p-type) 2.59 2.46 0.58 0.18 0.75 0.03

274

1016

1017

1018

1019

1020

Dopant Concentration (cm−3

)

0.0

0.5

1.0

1.5

2.0

Z1D

T

binary

bisectrix

[0112]

trigonal

[1011]

Figure F.6: Calculated Z1DT at 77 K as a function of Te dopant concentration Nd for 10 nmTe-doped Bi nanowires oriented along different directions: trigonal, binary, bisectrix, [1011],and [0112].

0 10 20 30 40 50 60diameter (nm)

0

1

2

3

4

Z1D

T

bisectrix

[1011]

[0112]

binary

trigonal

n−type

77 K

Figure F.7: Calculated Z1DT at 77 K as a function of Te dopant concentrations Nd for10 nm Te-doped Bi nanowires oriented along different directions: trigonal, binary, bisectrix,[1011], and [0112].

275

to the geometric limitations, the first attempt to study the temperature dependent prop-erties of Bi nanowires was made with a two-probe measurement. Although an absolutevalue of the resistivity cannot be derived through this two-probe method, the temperaturedependence can be examined by normalizing the resistance to that at a common tem-perature, e.g., R(300 K). Figure F.8 shows the temperature dependence of the resistivityR(T )/R(300 K) of Bi nanowire arrays of various wire diameters prepared by the vapordeposition process.

As shown in Fig. F.8, the temperature dependence of the resistance of Bi nanowiresis very different from that of bulk Bi, and is very sensitive to the wire diameter. At hightemperatures (T > 70 K), the resistance of all nanowire arrays shown in this figure increaseswith decreasing temperature. When T < 70 K, the resistance of the smaller diameter Binanowires (7 nm – 48 nm) in the semiconducting regime continues to increase with decreasingtemperature, while the resistance decreases with decreasing temperature for the nanowiresamples of larger diameters (70 nm and 200 nm) in the semimetallic regime.

The striking difference in the temperature dependence of the resistance between Binanowires arrays and bulk Bi can readily be explained qualitatively, and a simple physicalargument is given here in terms of the temperature dependence of the carrier density n(T )and mobility µ(T ). For both Bi nanowires and bulk Bi, n(T ) increases with increasingtemperature, while µ(T ) decreases. However, for Bi nanowires with larger wire diame-ters which are semimetallic (e.g., 70 nm and 200 nm), the increase in the carrier densitywith increasing temperature is much slower than that in the semiconducting wires withsmaller diameters (e.g., dW ≤ 48 nm), especially at low temperatures. For smaller diameternanowires (< 48 nm in Fig. F.8), the increase in n(T ) outweighs the decrease in µ(T ) whenthe temperature increases, and therefore the resistance drops. On the other hand, for bulkBi or Bi nanowire arrays with larger wire diameters (70 nm and 200 nm in Fig. F.8), thetemperature dependence of µ(T ) becomes more important due to the weaker T dependenceof n(T ) in the semimetallic regime, and therefore, the resistance increases with increasingtemperature for T < 100 K. The carrier density of the 7 nm semiconducting Bi nanowiresincreases by many orders of magnitude (∼ 108) when the temperature increases from 100 Kto 300 K, while that of the 70 nm nanowires also increases by about 16 times. However,since the mobility of bulk Bi decreases by a factor of 13 from 100 K to 300 K, the increasein n(T ) overwhelms that of µ(T ), and the resistance of Bi nanowires (7–70 nm) decreaseswith temperature for T > 100 K.

Based on the model for the electronic structure for Bi nanowires (see §F.3.3) and thetransport model (developed in §F.3.5 for the more general case of a doped Bi nanowiresystem), the normalized temperature dependent resistance R(T )/R(300 K) for 70 nm and36 nm Bi nanowires has been calculated, and the results are shown by the solid curves inFig. F.9, exhibiting trends consistent with the experimental results in Fig. F.8. Calculationsfor the wire diameters of 70 nm and 36 nm are particularly interesting, because these wiresrepresent two different types of Bi nanowires: semimetallic and semiconducting, respectively.In the modeling of Fig. F.9, some assumptions were made to take into account the discrep-ancies between an ideal 1D Bi quantum wire and a real Bi nanowire. First, in a perfectsingle crystalline Bi quantum wire, there is no scattering at the wire boundary because theelectron (or hole) wave-function and the local carrier density vanish at the wire boundary asa result of the assumed ideal infinite-potential interface. Within the Bi nanowire, possiblescattering mechanisms are electron-phonon and electron-electron interactions. However, in

276

Figure F.8: Temperature dependence of the normalized resistance for Bi nanowire arrays ofvarious wire diameters prepared by the vapor deposition method, in comparison with thecorresponding data for bulk Bi. The measurement of the resistance was made while the Binanowires were in their alumina templates using a two-probe measurement technique.

277

10 100T (K)

0

1

2

3

4

R(T

)/R

(270

K)

36 nm

70 nm

70 nm (polycrystalline)

Figure F.9: The calculated temperature dependence of the resistance for Bi nanowires of36 nm and 70 nm, using a semiclassical transport model.

a real Bi nanowire sample, the boundary conditions are far from ideal, and the energy bar-rier at the boundary is finite instead of infinite. Furthermore, real Bi nanowires may havea higher defect level in a thin layer at the boundary than in the interior of the nanowiresdue to the various surface conditions at the Bi/Al2O3 interface. Therefore, the electronswill experience substantial boundary scattering, due to the finite amplitude of the electronwave functions at the boundary. This boundary scattering effect has been observed in mag-netoresistance measurements (see §F.3.8). In addition to the scattering at the boundary,the electrons can also be scattered at grain boundaries within real Bi nanowire samples, forwhich a domain size on the order of the wire diameter has been observed. However, sincethe domains in Bi nanowires possess the same crystal orientation along the wire axis, thesmall-angle scattering at the grain boundary would be expected to be a minor scatteringmechanism in determining the transport properties of real Bi nanowires. Another strikingdifference between ideal Bi nanowires and real Bi nanowire samples is the uncontrolled im-purities which act as dopants in Bi nanowires. For ideal semiconducting Bi quantum wires,the carrier density should decay exponentially with decreasing T at low temperatures, andthe resistance should correspondingly increase dramatically. Instead, even for the 7 nm Binanowires, the measured resistance increases slowly and steadily with decreasing tempera-ture (see Fig. F.8). This effect can be attributed to the uncontrolled impurities in the Binanowires, which give rise to a finite carrier density at low temperatures. The uncontrolledimpurities will not only alter the carrier density, but will also decrease the carrier mobilityby ionized impurity scattering.

The effect of each scattering mechanism mentioned above can be characterized by a scat-tering time τ , and the total scattering time τtot in a real Bi nanowire can be approximatedby adding the scattering rates in accordance with Matthiessen’s rule

1

τtot(T )=

1

τbulk(T )+

1

τboundary+

1

τimp(T )(F.36)

in which τbulk is the total relaxation time in bulk Bi, and τboundary and τimp are the relaxation

278

times for boundary scattering (including wire and grain boundary scattering) and ionizedimpurity scattering, respectively. It should be noted that boundary scattering and ionizedimpurity scattering are only important at low temperatures (<100 K), and that phononscattering becomes the dominant scattering mechanism at higher temperatures (> 100 K).The relaxation time for ionized impurity scattering is approximately proportional to T 3/2,while boundary scattering is much less temperature dependent, and for simplicity τboundary

was assumed to be a constant, independent of temperature. Since the mobility µ is propor-tional to the relaxation time τ , the approximation used to find the mobility, considering allof these scattering effects, was:

1

µtot(T )=

1

µbulk(T )+

1

µboundary+

1

µimp(T )(F.37)

in which µbulk(T ) can be found in the literature, µboundary was assumed to be constant inT , and µimp ∼ T 3/2. In the curve for the 70 nm Bi nanowires in Fig. F.9, the two mobilitiesµboundary and µimp were fitted to 50 m2V−1s−1 and 1.0 × T 3/2 m2V−1s−1, respectively. Asfor the 36 nm Bi nanowires, the mobility terms were fit to µboundary ' 33 m2V−1s−1 andµimp ' 0.2 × T 3/2 m2V−1s−1, and the carrier density, due to the presence of uncontrolledimpurities, was fitted to Nimp ' 5 × 1016 cm−3, which amounts to less than 100 impurityatoms in the Bi nanowires per 1 µm in length. However, since the 70 nm Bi nanowires aresemimetallic at low temperatures, the carrier density contribution due to the small amountof uncontrolled impurities per se has an insignificant effect, and this effect is neglected inthe modeling, taking account only of the effect of these uncontrolled impurities on scat-tering carriers. We also note that µboundary is smaller for the 36 nm Bi nanowires thanfor the 70 nm nanowires, due to their smaller wire diameter. As the wire diameter de-creases, the contribution of the µimp term decreases more rapidly that the µboundary term.Regarding the main contribution of the last two terms to Eq. (F.37), the term µ−1

boundary

dominates over µ−1imp for 77 K and above, and the µ−1

imp term is relatively larger for the smalldiameter nanowires. The normalized resistance R(T )/R(300 K) curves in Fig. F.9 showgeneral trends for the temperature dependence of the normalized resistance of the 36 nmand 70 nm Bi nanowires, consistent with the experimental results in Fig. F.8 for the actualnanowire arrays, showing strong evidence that the different temperature dependences ofR(T )/R(270 K) for Bi nanowires with different wire diameters are predominantly due tothe quantum confinement-induced semimetal-semiconductor transition, which occurs whenthe wire diameter in Fig. F.8 decreases below 50 nm.

The effect of crystal quality can be accounted for in the same transport model by thevalue of µboundary in Eq. (F.37). Instead of a non-monotonic behavior for semimetallic Binanowires as shown in Fig. F.8, R(T ) is predicted to show a monotonic T dependence ata higher defect level. The dashed curve in Fig. F.9 shows the calculated R(T )/R(300 K)for 70-nm wires with increased boundary scattering (µboundary ' 6 m2V−1s−1), exhibitinga monotonic T dependence, similar to that of Bi nanowires prepared by electrochemicaldeposition which is likely to produce polycrystalline nanowires. Generally speaking, forsamples with many grain boundaries the 1/τboundary term is large, leading to a qualitativelydifferent temperature dependence and a lower overall mobility.

The differences in the slopes of the temperature dependence of the low temperatureresistance (T < 10 K) also provide experimental evidence for the semimetal [large nanowirediameter and (∂R/∂T ) > 0] to semiconductor [small nanowire diameter and (∂R/∂T ) < 0]

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Figure F.10: The temperature dependence of the zero-field resistance for Bi nanowires ofdifferent diameters, normalized to the resistance at 10 K.

transition in Bi nanowires, as shown in Fig. F.10.

F.3.8 Magnetoresistance of Bi Nanowires

Because of the inherent one-dimensional geometry of nanowires, certain conventional mea-surements, such as the Hall effect, which are traditionally carried out to determine thecarrier density cannot be performed. Magneto-oscillatory effects cannot be used in manycases to determine the Fermi energy because of wire boundary scattering (which makes itdifficult to satisfy ωcτ À 1), and optical measurements on the Bi/anodic alumina samplesto determine the plasma frequency are largely dominated by contributions from the hostalumina template, and even single nanowire measurements of the absolute resistivity arequite challenging. Therefore, determining the effects of doping and annealing Bi nanowiresoften cannot be assessed by conventional means.

Magnetoresistance (MR) measurements provide an informative technique for charac-terizing Bi nanowires because these measurements yield a great deal of information aboutelectron scattering from wire boundaries, the effects of doping and annealing on scattering,and localization effects in the nanowires.

Figure F.11 shows the longitudinal magnetoresistance (B parallel to the wire axis) for65 nm and 109 nm diameter Bi nanowire samples at 2 K. In the low field regime, the MRincreases with B (positive MR), up to some peak value, Bm, beyond which the MR becomesa decreasing function of B (negative MR). This behavior is typical of the longitudinal MRof Bi nanowires in the diameter range 45 nm to 200 nm, and can be understood on thebasis of the classical size effect of the nanowire. The MR of wires with diameters smallerthan 40 nm shows a strong dependence on B. The peak position Bm moves to lower Bfield values as the wire diameter increases, as shown in Fig. F.11(b,c) where Bm is seen tovary linearly with 1/dW . The application of a longitudinal magnetic field produces helicalmotion of the electrons along the wire, and above some critical field, approximately Bm,

280

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 1 2 3 4 5B (T)

∆R

(B)/

R(0

)

65-nm

109-nm

(b.)

(c.)

(a.)

Figure F.11: (a) Longitudinal magnetoresistance, ∆R(B)/R(0), at 2 K as a function of B forBi nanowire arrays with diameters 65 and 109 nm before thermal annealing. (b) The peakposition Bm as a function of temperature for the 109 nm diameter Bi nanowire array. (c)The peak position Bm of the longitudinal MR at 2 K as a function of 1/dW , the reciprocalof the nanowire diameter.

the radius of the helical motion will become smaller than the radius of the wire, causing adecrease in the wire boundary scattering, and giving rise to a negative magnetoresistance∂R/∂B < 0. However, for low fields B ≤ Bm, the magnetic field deflects the electronscausing increased scattering with the wire boundary, thereby giving rise to an increase inresistance or a positive magnetoresistance, which is common to most crystalline solids. Thecondition for Bm is given by Bm ∼ 2chkF /edW where kF is the wave vector at the Fermienergy. In summary, for B ≤ Bm the cyclotron radius is larger than the wire radius andwe have a positive MR, while for B ≥ Bm, the cyclotron radius is smaller than the wireradius, and we have a negative MR. This phenomenon, called the classical size effect for themagnetoresistance, provides much insight into the scattering of electrons in Bi nanowires.

The peak position, Bm, is found to increase linearly with temperature in the range 2 to100 K, as shown in Fig. F.11(b,c). As T is increased, phonon scattering becomes importantand therefore a higher magnetic field is required to reduce the resistivity associated withboundary scattering sufficiently to change the sign of the MR. Likewise increasing the grainboundary scattering also increases the value of Bm at a given T and wire diameter. Appli-cation of a transverse magnetic field does not show significant reduction in wire boundaryscattering, and therefore the transverse MR is always positive.

Thermal annealing of undoped Bi nanowire samples causes a significant decrease inthe magnitude of the magnetoresistance as well as a decrease in the peak position, Bm.This behavior indicates that prior to annealing, the scattering at defects and impurities isdominant over scattering at the wire boundary, even at low temperature (2 K). The observeddecrease in MR upon annealing indicates that the Bi nanowires become purer after thermal

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treatment, as one would expect.

Bi nanowires doped with Te have been fabricated and characterized, as discussed in§F.3.4. The longitudinal MR of Te-doped samples show no peak in the MR (as can beseen in Fig. F.11 for undoped samples), and instead, the longitudinal MR of Te-dopedsamples is found to be a monotonically increasing function of magnetic field (positive MR)in the magnetic field range 0 ≤ B ≤ 5.4 T at 2 K. The disappearance of the negative MR isattributed to a change in the dominant scattering mechanism from wire boundary scattering(which can be reduced by applying a B field) to magnetic field-independent ionized impurityscattering from the Te dopant ions.

Annealing the Te-doped samples yields MR behavior that is in striking contrast to thatof the undoped samples described above. Upon annealing, an increase in the MR of theTe-doped samples is observed. This indicates that the dopants are being pushed out ofthe nanowire to the wire boundary, thereby increasing the role of boundary scattering anddecreasing the role of charged impurity scattering.

For Te doped samples with dW < 40 nm, the longitudinal MR monotonically increasesas B2 and shows no peak, indicating that B, in the measured range (up to 5.4 T), is too lowto reduce the cyclotron radius below the wire radius. Consequently, increasing the magneticfield in this field range always leads to increased boundary scattering.

In addition to the longitudinal magnetoresistance measurements, transverse magnetore-sistance measurements (B perpendicular to the wire axis) have also been performed on Binanowire array samples, where a monotonically increasing B2 dependence over the entirerange 0 ≤ B ≤ 5.5 T is found for all Bi nanowires studied thus far. This is as expected,since the wire boundary scattering cannot be reduced by a magnetic field perpendicular tothe wire axis. The negative MR observed for the Bi nanowire arrays above Bm shows thatwire boundary scattering is a dominant scattering process for the longitudinal magnetore-sistance, thereby establishing that the mean free path is larger than the wire diameter andthat the Bi nanowires have high crystal quality.

Also encouraging for thermoelectric applications are the results on the high-field classi-cal size effect in the longitudinal magnetoresistance, showing that the defect and impuritylevels in the nanowires are sufficiently low so that the wire diameter is comparable to orsmaller than the carrier mean free path, and ballistic transport can occur in the nanowiresin a high longitudinal magnetic field. The ability of the electronic structure and transportmodels for Bi nanowires to account for the dependence of the classical size effect in the mag-netoresistance on temperature, magnetic field, nanowire diameter and annealing conditionsis important for predicting the behavior of Bi nanowires in the smaller diameter range, wellbelow 10 nm, where enhancement in Z1DT is expected.

By applying a magnetic field, a transition from a 1D localized system, which is character-istic of low magnetic fields, to a 3D localized system can be induced as the magnetic field isincreased. The effect of this transition can be seen in Fig. F.12, where the longitudinal mag-netoresistance is plotted for Bi nanowire arrays of various nanowire diameters in the range28 to 70 nm for T < 5 K. In these curves, a subtle step-like feature is seen at low magneticfields, and this feature is independent of temperature and of the orientation of the magneticfield, and depends only on wire diameter. The corresponding transverse magnetoresistancecurves, also show a step at the same magnetic field strengths. The lack of dependence ofthe magnetic field of the step on temperature and magnetic field orientation indicates thatthe phenomenon is not related to the effective masses, which are highly anisotropic in Bi,

282

but rather is related to the magnetic field length, LH = (h/eB)1/2, which is the spatialextent of the wave function of electrons in the lowest Landau level, and LH is independentof the effective mass. Setting LH(Bc) equal to the diameter dW of the nanowire, definesa critical magnetic field strength, Bc, below which the carrier wavefunction is confined bythe nanowire boundary (the 1D localization regime), and above which the wavefunction isconfined by the magnetic field (the 3D localization regime). This calculated field strength,Bc, is indicated in Fig. F.12 by vertical lines for the appropriate nanowire diameters, andthese calculated Bc values provide a good fit to the step-like like features in the MR curvesshown in Fig. F.12. The physical basis for this phenomenon is associated with localizationof a single magnetic flux quantum within the nanowire diameter.

The last magnetic field characterization technique discussed in this section is the Shubnikov–de Haas (SdH) quantum oscillatory effect. SdH oscillations, in principle, provide the mostdirect measurement of the Fermi energy and carrier density for the Bi nanowire system.However, in order to observe SdH oscillations, the magnetic field is applied parallel to thenanowire axis, and the electrons must complete at least one cyclotron orbit without beingscattered. Thus the cyclotron radius must be smaller than the wire radius and the meanfree path to observe the SdH effect. SdH oscillations occur when the quantized Landaulevels pass through the Fermi energy as the magnetic field is increased. By determining theperiod of the SdH oscillation (periodic in 1/B), the position of the Fermi energy can bedetermined by the relation

1

∆(1/B)=

mcEeF

hq[1 + Ee

F /Eg] (F.38)

where ∆(1/B) denotes the SdH period, mc is the cyclotron mass, EeF is the electron Fermi

level and Eg is the L-point gap, where non-parabolic effects are explicitly considered forelectrons, while for the holes, the non-parabolic term Eh

F /Eg can be neglected. Figure F.13shows SdH oscillations reported for an undoped Bi nanowire sample with a 200 nm wirediameter, which was found to be slightly n-type, due to uncontrolled impurities. Alsomeasurements of SdH oscillations were made on Te-doped 200 nm diameter Bi nanowires,also showing two different periods at 14.6 T−1 (identified with light mass electron orbits)and at 19.5 T−1 (identified with heavy mass electron orbits).

F.3.9 Seebeck Coefficient of Bi Nanowires

Thus far, there have been very few measurements on the Seebeck coefficient of Bi nanowires,though the recent achievement of reliable measurements on 200 nm Bi nanowire arrays isencouraging, and corroborates extensive prior studies of the thermoelectric properties ofbulk single crystals. Improvement in the measurement technique for the Seebeck coefficientof Bi nanowires is still needed to extend the measurements to the smaller nanowire diametersof interest for possible thermoelectric applications.

The Seebeck coefficient, unlike the resistivity, is intrinsically independent of sample sizeand the number of nanowires contributing to the signal, because S depends on the ratioof L(1)/L(0), on temperature and is expected to depend on wire diameter. The aluminatemplate containing an array of Bi nanowires therefore provides a convenient package formeasuring the Seebeck coefficient of Bi nanowires.

Two techniques for measuring the Seebeck coefficient of Bi nanowire arrays are describedin the literature, one using a differential thermocouple arrangement, and another, in which

283

Figure F.12: Longitudinal magnetoresistance as a function of magnetic field for Bi nanowireswith 28, 36, 48 and 70 nm diameters. The vertical bars indicate the critical magnetic fieldBc at which the magnetic length LH = (h/eB)1/2 equals the nanowire diameter.

284

Figure F.13: (a) The oscillatory magnetoresistance for the magnetic field parallel to thenanowire axis of an array of parallel undoped Bi nanowires 200 nm in diameter embeddedin an anodic alumina template after the background MR has been subtracted. (b) Fouriertransform of the oscillatory part of the magnetoresistance, showing two well-defined SdHperiods, the 4.2 T−1 period being identified with the heavy electron cyclotron orbit andanother period at 9.25 T−1 identified with T -point holes.

285

Figure F.14: The temperature dependence of the Seebeck coefficient of two undoped Binanowire arrays and one Te-doped Bi nanowire array. All three samples have Bi nanowirediameters of 200 nm. The results for the Seebeck coefficient for the Bi nanowires are com-pared to that of bulk Bi along the bisectrix direction indicated by the dashed curve.

the thermocouples are mounted in direct electrical contact with the sample to measure thetemperature difference, ∆T , across the Bi nanowire sample. The thickness of the thermo-couples used in both measurements was 12.5 µm, which is comparable to the 50 µm samplethickness. Because of the relatively large thickness of the thermocouples, the measured ∆Tis actually the temperature difference across both the sample and the thermocouples, andtherefore, the measured ∆T overestimates the true sample ∆T . Thus, the measured See-beck coefficient will be a lower limit of the true Seebeck coefficient of the sample. Despitethe small thickness of the samples, large ∆T ’s (in excess of 10 K ) are achievable becauseof the low thermal conductivity of the alumina template (∼1.7 W/mK).

The Seebeck coefficient of bulk Bi is low (−50 to −100 µV/K) because of the presence ofboth electrons and holes, whose contributions to S tend to cancel each other. Figures F.14and F.15 show the measured Seebeck coefficients of 200 nm Bi nanowires and 40 nm Binanowires, respectively. One immediately notices the low magnitudes for these Seebeckcoefficients. As mentioned above, the measurement technique underestimates the Seebeckcoefficient. For the 200 nm nanowires, the band overlap is not expected to deviate from the

286

0 100 200 300Temperature (K)

−30

−20

−10

0

S(µ

V/K

)

0.075% Te doped0.15% Te doped

Figure F.15: The temperature dependence of the Seebeck coefficient of two Te-doped Binanowire 40 nm diameter arrays with different doping concentrations.

bulk value and therefore 200 nm Bi nanowires contain approximately the same number ofholes and electrons as bulk Bi. We thus attribute the low values of S to the same cancella-tion between electron and hole contributions that occurs in bulk Bi and the underestimateof the measurement technique for measuring S. However for the 40 nm diameter wires,the band structure is expected to be significantly different; in particular, we expect theband overlap between the valence and conduction bands found in bulk Bi to be absent inthese nanowires which instead have a small bandgap, appropriate to the semiconductingstate, thus potentially providing a higher Seebeck coefficient. However, as shown by thecalculations presented in Fig. F.16, S is large when the chemical potential is near the bandedge, but S is small when the chemical potential is far from the band edge. The chemicalpotential can be varied by adding dopants which increase the electron (or hole) concentra-tion. The presence of Te dopants in Bi nanowires moves the chemical potential to lie withinthe conduction band, thus explaining the low measured values of the Seebeck coefficient.The theoretical optimum Z1DT is predicted to lie close to the conduction band edge (forn-type) as indicated in Fig. F.16. The challenge that now presents itself is how to controlthe chemical potential ζ sensitively enough to optimize the thermoelectric properties, andhow to measure ζ precisely enough for the optimization of Z1DT .

¿From Figs. F.14 and F.15 we notice a striking difference in the effect of doping fornanowires of different diameters (40 and 200 nm). Upon the addition of Te dopant, themagnitude of S is increased for the 200 nm diameter samples, as shown in Fig. F.14. How-ever, the magnitude of the Seebeck coefficient is decreased upon doping of the nanowireswith diameter 40 nm, as shown in Fig. F.15. These results appear to be in contrast witheach other. However, as we look at the difference in the band structure of these two sampleswith different diameters, the reasons for this contrasting behavior become clear. For the200 nm diameter nanowires, there is essentially no shift in the band edge energy relative tobulk bismuth. Therefore, at T = 77 K (for example), the nanowires are semimetallic with

287

−100 −50 0 50 100Chemical Potential (meV)

−200

−100

0

100

200

See

beck

Coe

ffici

ent (

µV/K

)

Conduction Band EdgeValence Band Edge

Figure F.16: The calculated Seebeck coefficient for a 40 nm diameter Bi nanowire withtransport along the [0112] direction at 77 K, considering non-parabolic dispersion relationsfor the conduction band and a circular wire cross-section.

a band overlap of 38 meV. S is calculated as a weighted sum of contributions from the holeand electron bands (Se and Sh), weighted by the hole and electron conductivities (σe andσh). The addition of Te dopant causes an increase in the electron conductivity and hencean increase in the magnitude of the negative Seebeck coefficient, since electrons dominateover holes in the transport phenomena of Bi nanowires. For the 40 nm diameter nanowires,however, the band edges are shifted appreciably due to quantum confinement effects. The40 nm nanowire is predicted to be semiconducting at low temperatures, with a calculatedbandgap of about 8 meV at 77 K. Figure F.16 shows the calculated Seebeck coefficient for40 nm diameter Bi nanowires oriented along the [0112] direction (the same orientation asthe wires in the sample of Fig. F.15), plotted as a function of the chemical potential. Alsoindicated on the figure are the conduction and valence band edges. It should be clear fromthis graph that the S of a lightly doped sample (where the chemical potential is expectedto lie just inside the conduction band) is larger in magnitude than the S of a heavily dopedsample (where the chemical potential lies further in the conduction band), thus explain-ing the contrasting behavior that is observed for the 200 nm and 40 nm diameter nanowiresamples.

F.4 Summary

In this appendix the predictions of an enhancement in the thermoelectric figure of merit ofBi quantum wires relative to their corresponding bulk counterparts, as well as the presentstate of experimental confirmation of these predictions are reviewed. Bismuth nanowires,offer significant promise for practical applications, because they can be self assembled andare predicted to have desirable thermoelectric properties when they have wire diametersin the 5–10 nm range. Though temperature-dependent resistance measurements have been

288

carried out for Bi nanowires in this diameter range, reliable thermoelectric measurementshave not yet been reported.

The introduction of low dimensional concepts into the field of thermoelectricity hasstimulated new approaches to thinking about better thermoelectric materials, new strategiesfor achieving higher Z3DT , and new applications areas for thermoelectrics, such as thermalmanagement of integrated circuits. It would be fair to say that the introduction of low-dimensional concepts into the thermoelectrics field has injected a large increase in interestand attention to thermoelectric materials and phenomena by the more general scientificcommunity. It is, however, too soon to assess the eventual impact of these low-dimensionalconcepts on the eventual use of thermoelectricity for practical applications.

289

Appendix G

Ion Implantation and Rutherford

Backscattering Spectroscopy

References:

• S.T. Picraux, Physics Today, November (1984), p. 38.

• S.T. Picraux and P.S. Peercy, Scientific American, March (1985), p. 102.

• J.W. Mayer, L. Eriksson and J.A. Davies, Ion Implantation of Semiconductors, Stan-ford University Press (1970).

• G. Carter and W.A. Grant, Ion Implantation of Semiconductors, Edward ArnoldPublishers (1976).

• J.K. Hirvonen, Ion Implantation, Treatise on Materials Science and Technology, Vol.18, Academic Press (1980).

• W.K. Chu, J.W. Mayer and M.A. Nicolet, Backscattering Spectrometry (AcademicPress, 1978)

G.1 Introduction to the Technique

Ions of all energies incident on a solid influence its materials properties. Ions are used inmany different ways in research and technology. We here review some of the physics of theinteraction of ion beams with solids and some of the uses of ion beams in semiconductors.Some useful reviews are listed above.

We start by noting that the interaction of ion beams with solids depends on the energyof the incident ion. A directed low energy ion (∼ 10–100 eV) comes to rest at or near thesurface of a solid, possibly growing into a registered epitaxial layer upon annealing (Fig.G.1). A 1–keV heavy ion beam is the essential component in the sputtering of surfaces.For this application, a large fraction of the incident energy is transferred to the atoms ofthe solid resulting in the ejection of surface atoms into the vacuum. The surface is left ina disordered state. Sputtering is used for removal of material from a sample surface onan almost layer–by–layer basis. It is used both in semiconductor device fabrication, ion

290

etching or ion milling, and more generally in materials analysis, through depth profiling.The sputtered atoms can also be used as a source for the sputter deposition techniquediscussed in connection with the growth of superlattices.

At higher energies, ∼ 100–300 keV, energetic ions are used as a source of atoms to modifythe properties of materials. Low concentrations, ¿ 0.1 atomic percent, of implanted atomsare used to change and control the electrical properties of semiconductors. The implantedatom comes to rest ∼ 1000A below the surface in a region of disorder created by the passageof the implanted ion. The electrical properties of the implanted layer depend on the speciesand concentration of impurities, the lattice position of the impurities and the amount oflattice disorder that is created.

Lattice site and lattice disorder as well as epitaxial layer formation can readily be ana-lyzed by the channeling of high energy light ions (such as H+ and He+), using the Rutherfordbackscattering technique. In this case MeV ions are used because they penetrate deeplyinto the crystal (microns) without substantially perturbing the lattice. This is an attractiveion energy regime because scattering cross sections, flux distributions in the crystal, and therate of energy loss are quantitatively established. Particle–solid interactions in the rangefrom about 0.1 MeV to 5 MeV are understood and one can use this well–characterized toolfor investigations of solid state phenomena. Outside of this range many of the conceptsdiscussed below remain valid: however, the use of MeV ions is favored in solid state charac-terization applications because of both experimental convenience and the ability to probeboth surface and bulk properties.

In § 6.2 and § 6.3, we will review the two last energy regimes, namely ion implantationin the 100 keV region and ion beam analysis (Rutherford backscattering and channeling) byMeV light mass particles (H+, He+). We start by describing the most important features ofion implantation. Then we will review a two atom collision in order to introduce the basicatomic scattering concepts needed to describe the slowing down of ions in a solid. Thenwe will state the results of the LSS theory (J. Lindhard, M. Scharff and H. Schiøtt, Mat.

Fys. Medd. Dan. Vid. Selsk. 33, No. 14 (1963)) which is the most successful basic theoryfor describing the distribution of ion positions and the radiation–induced disorder in theimplanted sample. A number of improvements to this model have been made in the lasttwo decades and are used for current applications. We will conclude this introduction toion implantation with a discussion of the lattice damage caused by the slowing down of theenergetic ions in the solid.

We then go on and describe the use of a beam of energetic (1–2 MeV) light massparticles (H+, He+) to study material properties in the near surface region (< µm) ofa solid. We will then see how to use Rutherford backscattering spectrometry (RBS) todetermine the stoichiometry of a sample composed of multiple chemical species and thedepth distribution of implanted ions. Furthermore, we see that with single–crystal targets,the effect of channeling also allows investigation of the crystalline perfection of the sampleas well as the lattice location of the implanted atomic species. Finally, we will review someexamples of the modification of material properties by ion implantation, including the useof ion beam analysis in the study of these materials modifications.

291

G.2 Ion Implantation

Ion implantation is an important technique for introducing impurity atoms in a controlledway, thus leading to the synthesis of new classes of materials, including metastable materials.The technique is important in the semiconductor industry for making p–n junctions by, forexample, implanting n–type impurities into a p–type host material. From a materials sciencepoint of view, ion implantation allows essentially any element of the periodic table to beintroduced into the near surface region of essentially any host material, with quantitativecontrol over the depth and composition profile by proper choice of ion energy and fluence.More generally, through ion implantation, materials with increased strength and corrosionresistance or other desirable properties can be synthesized.

A schematic diagram of an ion implanter is shown in Fig. G.2. In this diagram the“target” is the sample that is being implanted. In the implantation process, ions of energy Eand beam current ib are incident on a sample surface and come to rest at some characteristicdistance Rp with a Gaussian distribution of half width at half maximum ∆Rp. Typicalvalues of the implantation parameters are: ion energies E ∼ 100 keV, beam currents ib ∼50 µA, penetration depths Rp ∼ 1000 A and half-widths ∆Rp ∼ 300 A (see Fig. G.3). Inthe implanted regions, implant concentrations of 10−3 to 10−5 relative to the host materialsare typical. In special cases, local concentrations as high as 20 at.% (atomic percent) ofimplants have been achieved.

Some characteristic features of ion implantation are the following:

1. The ions characteristically only penetrate the host material to a depth Rp ≤ lµm.Thus ion implantation is a near surface phenomena. To achieve a large percentage ofimpurity ions in the host, the host material must be thin (comparable to Rp), andhigh fluences of implants must be used (φ > l016/cm2).

2. The depth profile of the implanted ions (Rp) is controlled by the ion energy. Theimpurity content is controlled by the ion fluence φ.

3. Implantation is a non–equilibrium process. Therefore there are no solubility limitson the introduction of dopants. With ion implantation one can thus introduce highconcentrations of dopants, exceeding the normal solubility limits. For this reason ionimplantation permits the synthesis of metastable materials.

4. The implantation process is highly directional with little lateral spread. Thus it ispossible to implant materials according to prescribed patterns using masks. Implan-tation proceeds in the regions where the masks are not present. An application of thistechnology is to the ion implantation of polymers to make photoresists with sharpboundaries. Both positive and negative photoresists can be prepared using ion im-plantation, depending on the choice of the polymer. These masks are widely used inthe semiconductor industry.

5. The diffusion process is commonly used for the introduction of impurities into semi-conductors. Efficient diffusion occurs at high temperatures. With ion implantation,impurities can be introduced at much lower temperatures, as for example room tem-perature, which is a major convenience to the semiconductor industry.

292

6. The versatility of ion implantation is another important characteristic. With the sameimplanter, a large number of different implants can be introduced by merely changingthe ion source. The technique is readily automated, and thus is amenable for useby technicians in the semiconductor industry. The implanted atoms are introducedin an atomically dispersed fashion, which is also desirable. Furthermore, no oxide orinterfacial barriers are formed in the implantation process.

7. The maximum concentration of ions that can be introduced is limited. As the im-plantation process proceeds, the incident ions participate in both implantation intothe bulk and the sputtering of atoms off the surface. Sputtering occurs because thesurface atoms receive sufficient energy to escape from the surface during the collisionprocess. The dynamic equilibrium between the sputtering and implantation processeslimits the maximum concentration of the implanted species that can be achieved.Sputtering causes the surface to recede slowly during implantation.

8. Implantation causes radiation damage. For many applications, this radiation damageis undesirable. To reduce the radiation damage, the implantation can be carriedout at elevated temperatures or the materials can be annealed after implantation.In practice, the elevated temperatures used for implantation or for post–implantationannealing are much lower than typical temperatures used for the diffusion of impuritiesinto semiconductors.

A variety of techniques are used to characterize the implanted alloy. Ion backscatteringof light ions at higher energies (e.g., 2 MeV He+) is used to determine the compositionversus depth with ∼ 10nm depth resolution. Depth profiling by sputtering in combinationwith Auger or secondary ion mass spectroscopy is also used. Lateral resolution is providedby analytical transmission electron microscopy. Electron microscopy, glancing angle x–rayanalysis and ion channeling in single crystals provide detailed information on the localatomic structure of the alloys formed. Ion backscattering and channeling are discussed inSection 6.3.

G.2.1 Basic Scattering Equations

An ion penetrating into a solid will lose its energy through the Coulomb interaction with theatoms in the target. This energy loss will determine the final penetration of the projectileinto the solid and the amount of disorder created in the lattice of the sample. When welook more closely into one of these collisions (Fig. G.4) we see that it is a very complicatedevent in which :

• The two nuclei with masses M1 and M2 and charges Z1 and Z2, respectively, repeleach other by a Coulomb interaction, screened by the respective electron clouds.

• Each electron is attracted by the two nuclei (again with the corresponding screening)and is repelled by all other electrons.

In addition, the target atom is bonded to its neighbors through bonds which usuallyinvolve its valence electrons. The collision is thus a many–body event described by acomplicated Hamiltonian. However experience has shown that very accurate solutions forthe trajectory of the nuclei can be obtained by making some simplifying assumptions. The

293

most important assumption for us, is that the interaction between the two atoms can beseparated into two components:

• ion (projectile)–nucleus (target) interaction

• ion (projectile)–electron (target) interaction.

Let us now use the very simple example of a collision between two masses M1 and M2 todetermine the relative importance of these two processes and to introduce the basic atomicscattering concepts required to describe the stopping of ions in a solid. Figure G.5 showsthe classical collision between the incident mass M1 and the target mass M2 which can bea target atom or a nearly free target electron.

By applying conservation of energy and momentum, the following relations can be de-rived (you will do it as homework).

• The energy transferred (T ) (Ref. H. Goldstein, Classical Mechanics, Academic Press(1950)) in the collision from the incident projectile M1 to the target particle M2 isgiven by

T = Tmax sin2(

Θ

2

)

(G.1)

where

Tmax = 4E0M1M2

(M1 + M2)2 (G.2)

is the maximum possible energy transfer from M1 to M2.

• The scattering angle of the projectile in the laboratory system of coordinates is givenby

cos θ =1 − (1 + M2/M1)(T/2E)

√

(1 − T/E)(G.3)

• The energies of the projectile before (E0) and after (E1) scattering are related by

E1 = k2E0 (G.4)

where the kinematic factor k is given by

k =

M1 cos θ ± (M2

2 − M12 sin2 θ)

1/2

M1 + M2

(G.5)

These relations are absolutely general no matter how complex the force between the twoparticles, so long as the force acts along the line joining the particles and the electron isnearly free so that the collision can be taken to be elastic. In reality, the collisions betweenthe projectile and the target electrons are inelastic because of the binding energy of theelectrons. The case of inelastic collisions will be considered in what follows.

Using Eqs. (G.2) and (G.3) and assuming either that M2 is of the same atomic speciesas the projectile (M2 = M1), or that M2 is a nearly free electron (M2 = me) we constructTable G.1.

With the help of the previous example, we can formulate a qualitative picture of theslowing down process of the incident energetic ions. As the incident ions penetrate into thesolid, they lose energy. There are two dominant mechanisms for this energy loss:

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Table G.1: Maximum energy trans-fer Tmax and scattering angle θ fornuclear (M2 = M1) and electronic(M2 = me) collisions.

“nuclear” collision “electronic” collision

Tmax Tmax ' E0 Tmax ' (4m/M1) E0

θ 0 < θ ≤ π/2 θ = 0◦

1. The interaction between the incident ion and the electrons of the host material. Thisinelastic scattering process gives rise to electronic energy loss.

2. The interaction between the incident ions and the nuclei of the host material. This isan elastic scattering process which gives rise to nuclear energy loss.

The ion–electron interaction (see Table G.1) induces small losses in the energy of theincoming ion as the electrons in the atom are excited to higher bound states or are ionized.These interactions do not produce significant deviations in the projectile trajectory. Incontrast, the ion–nucleus interaction results in both energy loss and significant deviationin the projectile trajectory. In the ion–nucleus interaction, the atoms of the host are alsosignificantly dislodged from their original positions giving rise to lattice defects, and thedeviations in the projectile trajectory will give rise to the lateral spread of the distributionof implanted species.

Let us now further develop our example of a “nuclear” collision. For a given interactionpotential V (r), each ion coming into the annular ring of area 2πpdp with energy E, willbe deflected through an angle θ where p is the impact parameter (see Fig. G.6). We defineT = E0 − E1 as the energy transfer from the incoming ion to the host and we define2πpdp = dσ as the differential cross section. When the ion moves a distance ∆x in the hostmaterial, it will interact with N∆x2πpdp atoms where N is the atom density of the host.

The energy ∆E lost by an ion traversing a distance ∆x will be

∆E = N∆x

∫

T 2πpdp (G.6)

so that as ∆x → 0, we have for the stopping power

dE

dx= N

∫

Tdσ (G.7)

where σ denotes the cross sectional area. We thus obtain for the stopping cross section E

E =1

N

dE

dx=

∫

Tdσ. (G.8)

The total stopping power is due to both electronic and nuclear processes

dE

dx=

(dE

dx

)

e+

(dE

dx

)

n= N(Ee + En) (G.9)

where N is the target density and Ee and En are the electronic and nuclear stopping crosssections, respectively. Likewise for the stopping cross section E we can write

E = Ee + En. (G.10)

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Table G.2: Typical Values of E1, E2,E3 for silicon.a

Ion E1(keV) E2(keV) E3(keV)

B 3 17 3000P 17 140 ∼ 3 × 104

As 73 800 > l05

Sb 180 2000 > 105

aSee Fig. G.7 for the definition of the notation.

From the energy loss we can obtain the ion range or penetration depth

R =

∫dE

dE/dx. (G.11)

Since we know the energy transferred to the lattice (including both phonon generationand displacements of the host ions), we can calculate the energy of the incoming ions as afunction of distance into the medium E(x).

At low energies of the projectile ion, nuclear stopping is dominant, while electron stop-ping dominates at high energies as shown in the characteristic stopping power curves ofFigs. G.7 and G.8. Note the three important energy parameters on the curves shown inFig. G.7: E1 is the energy where the nuclear stopping power is a maximum, E3 where theelectronic stopping power is a maximum, and E2 where the electronic and nuclear stoppingpowers are equal. As the atomic number of the ion increases for a fixed target, the scaleof E1, E2 and E3 increases. Also indicated on the diagram is the functional form of theenergy dependence of the stopping power in several of the regimes of interest. Typical val-ues of the parameters E1, E2 and E3 for various ions in silicon are given in Table G.2. Ionimplantation in semiconductors is usually done in the regime where nuclear energy loss isdominant. The region in Fig. G.7 where (dE/dx) ∼ 1/E corresponds to the regime wherelight ions like H+ and He+ have incident energies of 1–2 MeV and is therefore the region ofinterest for Rutherford backscattering and channeling phenomena.

G.2.2 Radiation Damage

The energy transferred from the projectile ion to the target atom is usually sufficient toresult in the breaking of a chemical bond and the permanent displacement of the targetatom from its original site (see Fig. G.9). The condition for this process is that the energytransfer per collision T is greater than the binding energy Ed.

Because of the high incident energy of the projectile ions, each incident ion can dislodgemultiple host ions. The damage profile for low dose implantation gives rise to isolatedregions of damage as shown in Fig. G.10.

As the fluence is increased, these damaged regions coalesce as shown in Fig. G.10. Thedamage profile also depends on the mass of the projectile ion, with heavy mass ions of agiven energy causing more local lattice damage as the ions come to rest. Since (dE/dx)n

increases as the energy decreases, more damage is caused as the ions are slowed down andcome to rest. The damage pattern is shown in Fig. G.11 schematically for light ions (suchas boron in silicon) and for heavy ions (such as antimony in silicon). Damage is causedboth by the incident ions and by the displaced energetic (knock-on) ions.

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A schematic diagram of the types of defects caused by ion implantation is shown inFig. G.12. Here we see the formation of vacancies and interstitials, Frenkel pairs (the pairformed by the Coulomb attraction of a vacancy and an interstitial). The formation ofmultiple vacancies leads to a depleted zone while multiple interstitials lead to ion crowding.

G.2.3 Applications of Ion Implantation

For the case of semiconductors, ion implantation is dominantly used for doping purposes,to create sharp p-n junctions in the near-surface region. To reduce radiation damage, im-plantation is sometimes done at elevated temperatures. Post implantation annealing is alsoused to reduce radiation damage, with elevated temperatures provided by furnaces, lasersor flash lamps. The ion implanted samples are characterized by a variety of experimentaltechniques for the implant depth profile, the lattice location of the implant, the residuallattice disorder subsequent to implantation and annealing, the electrical properties (Halleffect and conductivity) and the device performance.

A major limitation of ion implantation for modifying metal surfaces has been the shallowdepth of implantation. In addition, the sputtering of atoms from the surface sets a maximumconcentration of elements which can be added to a solid, typically ∼ 20 to 40 at.%. To formthicker layers and higher concentrations, combined processes involving ion implantation andfilm deposition are being investigated. Intense ion beams are directed at the solid to bringabout alloying while other elements are simultaneously brought to the surface, for exampleby sputter deposition, vapor deposition or the introduction of reactive gases. One processof interest is ion beam mixing, where thin films are deposited onto the surface first andthen bombarded with ions. The dense collision cascades of the ions induce atomic–scalemixing between elements. Ion beam mixing is also a valuable tool to study metastablephase formation.

With regard to polymers, ion implantation can enhance the electrical conductivity bymany orders of magnitude, as is for example observed (see Fig. G.13) for ion implanted poly-acrylonitrile (PAN, a graphite fiber precursor). Some of the attendant property changesof polymers due to ion implantation include cross-linking and scission of polymer chains,gas evolution as volatile species are released from polymer chains and free radical forma-tion when vacancies or interstitials are formed. Implantation produces solubility changesin polymers and therefore can be utilized for the patterning of resists for semiconductormask applications. For the positive resists, implantation enhances the solubility, while fornegative resists, the solubility is reduced. The high spatial resolution of the ion beamsmakes ion beam lithography a promising technique for sub–micron patterning applications.For selected polymers (such as PAN), implantation can result in transforming a good insu-lator into a conducting material with an increase in conductivity by more than 10 ordersof magnitude upon irradiation. Thermoelectric power measurements on various implantedpolymers show that implantation can yield either p–type or n–type conductors and in facta p–n junction has recently been made in a polymer through ion implantation (T. Wada,A. Takeno, M. Iwake, H. Sasabe, and Y. Kobayashi, J. Chem. Soc. Chem. Commun.,17, 1194 (1985)). The temperature dependence of the conductivity for many implanted

polymers is of the form σ = σo exp(T0/T )1/2

which is also the relation characteristic of theone–dimensional hopping conductivity model for disordered materials. Also of interest is

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the long term chemical stability of implanted polymers.

Due to recent developments of high brightness ion sources, focused ion beams to sub-micron dimensions can now be routinely produced, using ions from a liquid metal source.Potential applications of this technology are to ion beam lithography, including the possi-bility of maskless implantation doping of semiconductors. Instruments based on these ideasmay be developed in the future. The applications of ion implantation represent a rapidlygrowing field.

Instruments based on these ideas may be developed in the future. The applications ofion implantation represent a rapidly growing field.

Further discussion of the application of ion implantation to the preparation of metastablematerials is presented after the following sections on the characterization of ion implantedmaterials by ion backscattering and channeling.

G.3 Ion Backscattering

In Rutherford backscattering spectrometry (RBS), a beam of mono-energetic (1–2 MeV),collimated light mass ions (H+, He+) impinges (usually at near normal incidence) on a targetand the number and energy of the particles that are scattered backwards at a certain angleθ are monitored (as shown in Fig.G.14) to obtain information about the composition of thetarget (host species and impurities) as a function of depth. With the help of Fig.G.15 wewill review the fundamentals of the RBS analysis.

Particles scattered at the surface of the target will have the highest energy E upondetection. Here the energy of the backscattered ions E is given by the relation

E = k2E0 (G.12)

where

k =

M1 cos θ ± (M2

2 − M12 sin2 θ)

1/2

M1 + M2

(G.13)

as discussed in Section G.2. For a given mass species, the energy Es of particles scatteredfrom the surface corresponds to the edge of the spectrum (see Fig. G.15). In addition, thescattered energy depends through k on the mass of the scattering atom. Thus differentspecies will appear displaced on the energy scale of Fig. G.15, thereby allowing for theirchemical identification. We next show that the displacement along the energy scale fromthe surface contribution gives information about the depth where the backscattering tookplace. Thus the energy scale is effectively a depth scale.

The height H of the RBS spectrum corresponds to the number of detected particles ineach energy channel ∆E.

G.4 Channeling

If the probing beam is aligned nearly parallel to a close-packed row of atoms in a singlecrystal target, the particles in the beam will be steered by the potential field of the rows ofatoms, resulting in an undulatory motion in which the “channeled” ions will not approachthe atoms in the row to closer than 0.1–0.2 A. This is called the channeling effect (D.V.

298

Morgan, Channeling (Wiley, 1973)). Under this channeling condition, the probability oflarge angle scattering is greatly reduced. As a consequence, there will be a drastic reductionin the scattering yield from a channeled probing beam relative to the yield from a beamincident in a random direction (see Fig. G.16). Two characteristic parameters for channelingare the normalized minimum yield χmin = HA/H which is a measure of the crystallinityof the target, and the critical angle for channeling ψ1/2 (the halfwidth at half maximumintensity of the channeling resonance) which determines the degree of alignment requiredto observed the channeling effect.

The RBS–channeling technique is frequently used to study radiation–induced latticedisorder by measuring the fraction of atom sites where the channel is blocked.

In general, the channeled ions are steered by the rows of atoms in the crystal. Howeverif some portion of the crystal is disordered and lattice atoms are displaced so that theypartially block the channels, the ions directed along nominal channeling directions can nowhave a close collision with these displaced atoms, so that the resulting scattered yield will beincreased above that for an undisturbed channel. Furthermore, since the displaced atomsare of equal mass to those of the surrounding lattice, the increase in the yield occurs at aposition in the yield vs. energy spectrum corresponding to the depth at which the displacedatoms are located. The increase in the backscattering yield from a given depth will dependupon the number of displaced atoms, so the depth (or equivalently, the backscatteringenergy E) dependence of the yield, reflects the depth dependence of displaced atoms, andintegrations over the whole spectrum will give a measure of the total number of displacedatoms. This effect is shown schematically in Fig.G.17.

Another very useful application of the RBS–channeling technique is in the determina-tion of the location of foreign atoms in a host lattice. Since channeled ions cannot approachthe rows of atoms which form the channel closer than ∼ 0.1A, we can think of a “for-bidden region”, as a cylindrical region along each row of atoms with radius ∼ 0.1A, suchthat there are no collisions between the channeled particles and atoms located within theforbidden zone. In particular, if an impurity is located in a forbidden region it will notbe detected by the channeled probing beam. On the other hand, any target particle canbe detected by (i.e., will scatter off) probing particles from a beam which impinges in arandom direction. Thus, by comparing the impurity peak observed for channeling and ran-dom alignments, the fraction of impurities sitting in the forbidden region of a particularchannel (high symmetry crystallographic axis) can be determined. Repeating the procedurefor other crystallographic directions allows the identification of the lattice location of theimpurity atom in many cases.

Rutherford backscattering will not always reveal impurities embedded in a host matrix,in particular if the mass of the impurities is smaller than the mass of the host atoms. Insuch cases, ion induced x–rays and ion induced nuclear reactions are used as signatures forthe presence of the impurities inside the crystal, and the lattice location is derived from thechanges in yield of these processes for random and channeled impingement of the probingbeam.

Ion markers consist of a very thin layer of a guest atomic species are embedded inan otherwise uniform host material of a different species to establish reference distances.Backscattering spectra are taken before and after introduction of the marker. The RBSspectrum taken after insertion of the marker can be used as a reference for various applica-tions. Some examples where marker references are useful include:

299

• Estimation of surface sputtering by ion implantation. In this case, recession of thesurface from the reference position set by the marker (see Fig. G.18) can be measuredby RBS and can be analyzed to yield the implantation–induced surface sputtering.

• Estimation of surface material vaporized through laser annealing, rapid thermal an-nealing or laser melting of a surface.

• Estimation of the extent of ion beam mixing.

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Figure G.1: Schematic illustrat-ing the interactions of ion beamswith a single–crystal solid. Di-rected beams of ∼ 10 eV areused for film deposition and epi-taxial formation. Ion beams ofenergy ∼ 1 keV are employed insputtering applications; ∼ 100keV ions are used in ion implan-tation. Both the sputtering andimplantation processes damageand disorder the crystal. Higherenergy light ions are used for ionbeam analysis.

Figure G.2: Schematic Dia-gram of Ion Implanter.

301

Figure G.3: Typical ion im-plantation parameters.

Figure G.4: Penetration ofions into solids.

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Figure G.5: The upper figure definesthe scattering variables in a two–bodycollision. The projectile has mass M1

and an initial velocity v0, and an im-pact parameter, p, with the targetparticle. The projectile’s final angle ofdeflection is θ and its final velocity isv1. The target particle with mass, M2,recoils at an angle φ with velocity v2.The lower figure is the same scatter-ing event in the center–of–mass (CM)coordinates in which the total momen-

tum of the system is zero. The coor-dinate system moves with velocity vc

relative to the laboratory coordinates,and the angles of scatter and recoil areΘ and Φ in the center of mass system.

Figure G.6: Classical model for the collision of a projectile of energy E with a target atrest. The open circles denote the initial state of the projectile and target atoms, and thefull circles denote the two atoms after the collision.

303

Figure G.7: Nuclear and electronic energy loss (stopping power) vs. Energy.

Figure G.8: Nuclear and electronic energy loss (stopping power) vs. ε1/2 (reduced units ofLSS theory).

304

Figure G.9: Energy transfer from projectile ionto target atoms for a single scattering event forthe condition T > Ed where Ed is the bindingenergy and T is the energy transfer per collision.

Figure G.10: Schematic diagram showing therange of lattice damage for low dose and highdose implants.

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Figure G.11: Schematic diagram of the damage pattern for light ions and heavy ions in thesame target (silicon).

Figure G.12: Example of typical defects induced by ion implantation.

306

Figure G.13: Implantation inducedconductivity of a normally insulatingpolymer.

Figure G.14: In the RBS experiment, the scattering chamber where the analysis/experimentis actually performed contains the essential elements: the sample, the beam, the detector,and the vacuum pump.

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Figure G.15: Schematic diagram showing the energy distribution of ions back-scatteredfrom a Si sample (not aligned) which was implanted with As atoms.

Figure G.16: Schematic backscat-tering spectrum and angular yieldprofile.

308

Figure G.17: Schematic random andaligned spectra for MeV 4He ions in-cident on a crystal containing dis-order. The aligned spectrum for aperfect crystal without disorder isshown for comparison. The differ-ence (shaded portion) in the alignedspectra between disordered and per-fect crystals can be used to deter-mine the concentration ND(0) of dis-placed atoms at the surface.

Figure G.18: Schematic of the marker experiment which demonstrates surface recessionthrough surface sputtering.

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